Hyperspaces: Fundamentals and Recent Advances

HYPERSPACES

PURE

AND APPLIED

A Program of Monographs,

MATHEMATICS Textbooks,

EXECUTIVE

and Lecture Notes

EDITORS

Earl J. Taft Rutgers University New Brunswick, New Jersey

EDITORIAL M S. Baouendi Universi~ of California, San Diego Jane Cronin Rutgers Universi~ Jack K. Hale Georgia Institute of Technology

Zuhair Nashed University of Delaware Newark, Delaware

BOARD Anil Nerode Cornell Universify Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

Gian-Carlo Rota Massachusetts Institute of Technology

Marvin Marcus University of Calrlfornia, Santa Barbara

David L. Russell Virginia Polytechnic Institute and State University

W. S. Massey Yale University

Walter Schempp Universitat Siegen

Mark Teply University of Wisconsin, Milwaukee

MONOGRAPHS AND TEXTBOOKS PURE AND APPLIED MATHEMATICS

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. z:. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.

IN

K. Yano, Integral Formulas in Riemannian Geometry (I 970) S. Kobayashi, Hyperbolic Manifolds and Holomorphrc Mappings (I 970) V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; Littlewood, trans.) (1970) B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translatron ed.; K. Makowskr, trans.) (1971) L. harici er a/., Functional Analysis and Valuation Theory (I 971) S. S. Passman, Infinite Group Rings (1971) L. Domhoff, Group Representation Theory. Part A: Ordinary Representatron Theory. Part B: Modular Representation Theory (1971, 1972) W. Boofhby and G. L. Weiss, eds., Symmetric Spaces (1972) Y. Mafsushima, Differentiable Manifolds (E. T. Kobayashr, trans.) (1972) L. E. Ward, Jr., Topology (1972) A. Babakhanian, Cohomological Methods in Group Theory (1972) R. Gilmer, Multiplicative Ideal Theory (I 972) J. Yeh, Stochastic Processes and the Wiener Integral (I 973) J. Barros-Nero, Introduction to the Theory of Distributions (I 973) R. Larsen, Functional Analysis (1973) K. Yano and S. lshihara, Tangent and Cotangent Bundles (1973) C. Procesi, Rings with Polynomial Identities (1973) R. Hermann, Geometry, Physics, and Systems (I 973) N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) J. Dieudonnt!, Introduction to the Theory of Formal Groups (1973) /. Vaisman, Cohomologv and Differential Forms (I 973) B.-Y. Chen, Geometryof Submanifolds (1973) M. Marcus, Finite Dimensronal Multilinear Algebra (in two parts) (1973, 1975) R. Larsen, Banach Algebras (I 973) R. 0. Kujala and A. L. Vitter, eds., Value Distribution Theory: Part A; Part Deficit and Bezout Estimates by Wilhelm Stall (1973) K. 8. Stolarsky, Algebrarc Numbers and Diophantine Approxrmation (1974) A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) B. R. McDonald, Finite Rings with Identity (1974) J. Satake, Linear Algebra (S. Koh et al., trans.) (I 975) J. S. Go/an, Localization of Noncommutative Rings (1975) G. Klambauer, Mathematical Analysis (1975) M. K. Agosfon, Algebraic Topology (I 976) K. R. Goodearl, Ring Theory (1976) L. E. Mansfield, Linear Algebra with Geometric Applications (1976) N. J. Pullman, Matrix Theory and Its Applicatrons (1976) B. R. McDonald, Geometric Algebra Over Local Rings (I 976) C. W. Groetsch, Generalized Inverses of Linear Operators (1977) J. E. Kuczkowski and J. L. Gersfing, Abstract Algebra (1977) C. 0. Christenson and W. L. Voxman, Aspects of Topology (1977) M. Nagata, Field Theory (1977) R. L. Long, Algebraic Number Theory (1977) W. F. Pfeffer, Integrals and Measures (1977) R. L. Wheeden and A. Zygmund, Measure and Integral (I 977) J. H. Curtiss, Introduction to Functions of a Complex Variable (1978) K. Hrbacek and T. Jech, Introduction to Set Theory (I 978) W. S. Massey, Homology and Cohomology Theory (I 978) M. Marcus, Introduction to Modern Algebra (1978) E. C. Young, Vector and Tensor Analysis (I 978) S. 8. Nadler, Jr., Hvperspaces of Sets (I 978) S. K. Segal, Topics.in Group Kings (1978) A. C. M. van Roofi, Non-Archimedean Functional Analysis (1978) L. Corwin and R..Szczarba, Calculus in Vector Spaces (I 979) C. Sadosky, Interpolation of Operators and Singular Integrals (I 979)

A.

8:

54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109.

J. Cronin, Differential Equations (1980) C. W. Groefsch, Elements of Applicable Functional Analysis (1980) 1. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) H. /. Freedan, Deterministic Mathematical Models rn Population Ecoloav (1980) S. B. Chae, Lebesgue Integration (1980) C. S. Rees el al., Theory and Applications of Fourier Analysis (1981) L. Nachbin, Introduction to Functional Analysis (R. M. Aron, trans.) (1981) G. Orzech and M. Drzech, Plane Algebraic Curves (1981) R. Johnsonbaugh and W. E. F’faffenbefger, Foundations of Mathematical Analysrs (1981) W. L. Voxman and R. H. Goetschel, Advanced Calculus (1981) L. J. Corwin and R. H. Szczarba, Multivanable Calculus (? 982) V. /. Isfr~fescu, Introduction to Linear Operator Theory (1981) R. D. Jiirvinen, Finite and Infinite Dimensional Linear Spaces (1981) J. K. Beem andP. E. Ehrlich, Global Lorentzian Geometry (1981) D. L. Armacost, The Structure of Locally Compact Abelian Groups (I 981) J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute (1981) K. H. Kim, Boolean Matrix Theory and Applications (I 982) T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) D. B. Gauld, Differential Topology (I 982) R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) M. Carmeli, Statistical Theory and Random Matrices (1983) J. H. Carruth et a/., The Theory of Topological Semigroups (I 983) R. L. Faber, Differential Geometry and Relativity Theory (1983) S. Earner?, Polynomials and Linear Control Systems (1983) G. Karpilovsky, Commutative Group Algebras (I 983) F. Van Oystakyen and A. Vefschor&, Relative Invariants of Rings (1983) 1. Vaisman, A First Course in Differential Geometry (1984) G. W. Swan, Applications of Optimal Control Theory in Biomedicine (I 984) T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) T. Albu and C. NiQt&escu, Relative Finiteness in Module Theory (I 984) K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition (I 984) F. Van Oystaeyen and A. Verschoren, Relative lnvanants of Rings (1984) B. R. McDonald, Lrnear Algebra Over Commutative Rings (I 984) M. Namba, Geometry of Projective Algebraic Curves (1984) G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) M. R. Bremner et a/., Tables of Dominant Weight Multiplicities for Representations of Simple Lre Algebras (1985) A. E. Fekete, Real Linear Algebra (1985) S. B. Chae, Holomorphy and Calculus in Normed Spaces (I 985) A. J. Jerri, Introduction to Integral Equations with Applications (1985) G. Karpilovsky, Projective Representations of Finite Groups (I 985) L. Nariciand E. Beckenstein, Topological Vector Spaces (1985) J. Weeks, The Shape of Space (1985) P. R. Gnbik and K. 0. Kortanek, Extrernal Methods of Operations Research (1985) J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) G. D. Crown et a/., Abstract Algebra (1986) J. H. Carruth et a/., The Theory of Topological Semigroups, Volume 2 (1986) R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras (I 986) M. W. Jeter, Mathematical Programmrng (1986) M. Alfman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (I 986) A. Verschoren, Relative Invariants of Sheaves (1987) R. A. Usmani, Applied Lrnear Algebra (1987) P. B/ass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) J. A. Reneke et a/., Structured Hereditary Systems (1987) H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesics (1987) R. Harte, lnvertibility and Singularity for Bounded Linear Operators (1988)

110. 1 11. 112. 1 13. 1 14. 115. 116. 1 17.

1 18. 1 19. 120. 121.

122. 123.

124. 125. 126.

127. 128.

129. 130.

131. 132. 133.

134. 135. 136. 137. 138. 139.

140. 141. 142. 143. 144.

145. 146. 147.

148. 149. 150.

151 152. 153. 154. 155. 156.

157. 158. 159. 160. 161. 162. 163. 164.

G. S. Ladde ef al., Oscillation Theory of Differential Equations with Deviating Arguments (I 987) L. Dudkin et a/,, Iterative Aggregation Theory (1987) T. Okubo, Differential Geometry (1987) D. 1. Stancl and M. L. Stancl, Real Analysis with Point-Set Topology (1987) T. C. Gard, Introduction to Stochastic Differential Equations (19881 S. S. Abhyankar, Enumerative Combmatorics of Young Tableaux (19881 H. Strade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988) J. A. Huckaba, Commutative Rings with Zero Divisors (1988) W. D. Walks, Combinatorial Designs (19881 W. Wi@aw, Topological Fields (1988) G. Karpilovsky, Field Theory (I 988) S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989) W. Kozlowski, Modular Function Spaces (1988) f. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (I 989) M. Pave/, Fundamentals of Pattern Recognition (I 989) V. Lakshmikantham ef a/., Stability Analysis of Nonlinear Systems (1989) R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (19891 N. A. Wafson, Parabolic Equations on an Infinite Strip (1989) K. J. Hastings, Introduction to the Mathematics of Operations Research (1989) B. Fine, Algebraic Theory of the Bianchi Groups (1989) D. N. Dikranian era/., Topological Groups (1989) J. C. Morgan II, Point Set Theory (1990) P. Eiler and A. Witkowski, Problems in Mathematical Analysis (1990) H. J. Sussmann, Nonlinear Controllability and Optimal Control (I 990) J.-P. Florens e? a/., Elements of Bayesian Statistics (1990) N. She//, Topological Fields and Near Valuations (1990) B. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engrneers (1990) S. S. Ho//and, Jr., Applied Analysis by the Hilbert Space Method (1990) J. Okninski, Semigroup Algebras (I 9901 K. Zhu, Operator Theory in Function Spaces (I 990) G. 6. Price, An Introduction to Multicomplex Spaces and Functions (I 991) R. 8. Darsr, Introduction to Linear Programming (I 991) P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) T. Husain, Orthogonal Schauder Bases (1991) J. Foran, Fundamentals of Real Analysis (I 991) W. C. Brown, Matrices and Vector Spaces (1991 i M. M. Rao andZ. D. Ren, Theory of Orlicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structures of Rings (1991) C. Small, Arithmetic of Finite Fields (1991) K. Yano. Comolex Alaebraic Geometrv (1991) D. G. Hoffman et a/.,-Coding Theory (1991) . M. 0. Gonzdlez, Classical Complex Analvsis (I 992) M. 0. Gondlez; Complex Analysis (1992) L. W. Baggeff, Functional Analysis (I 992) M. Sniedovich, Dynamic Programming (1992) R. P. Agarwal, Difference Equations and Inequalities (I 992) C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis (1992) C. Swarfz, An Introduction to Functional Analysis (1992) S. B. Nadlef, Jr., Continuum Theory (1992) M. A. Al-Gwaiz, Theory of Distributions (19921 E. Pen-y, Geometry: Axiomatic Developments with Problem Solving (1992) E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modellinq in Science and Engineering (1992) A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analvsis (1992) A. Charlier et a/., Tensors and the Clifford Algebra (1992) P. Bile/ and T. Nadzieja, Problems and Examples in Differential Equations (1992)

165. 166. 167. 168. 169. 170. 171.

E. Hansen, Global Optimization Using Interval Analysis (1992) S. Guerre-De/abri&re, Classical Sequences in Banach Spaces (1992) Y. C. Wong, Introductory Theory of Topological Vector Spaces (1992) S. H. Kulkarni and B. V. Limaye, Real Function Algebras (1992) W. C. Brown, Matrices Over Commutative Rings (1993) J. Lousfau and M. Dillon. Linear Geometry with Computer Graphics (1993) W. V. Petryshyn. Approximation-Solvability of Nonlinear Functional and Differential Equations (1993) 172. E. C. Young, Vector and Tensor Analysis: Second Edition (1993) 173. T. A. Eick, Elementary Boundary Value Problems (I 993) 174. M. Pave/, Fundamentals of Pattern Recognition: Second Edition (1993) S. A. Albeverio er a/., Noncommutative Distributions (1993) 175. 176. W. I%/&, Complex Variables (1993) M. M. Rao, Conditional Measures and Applications (1993) 177. 178. A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (I 994) P. Neirraanmgki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems 179. (1994) Differential Equations: Introduction and Qualitative Theory, Second 180. J. Cronin, Edition (I 994) 181. S. Heikkil.+’ and V. Lakshmikanrham. Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) 182. X. Mao, Exponential Stability of Stochastic Differential Equations (1994) B. S. Thomson, Symmetric Properties of Real Functions (1994) 783. 184. J. E. Rubio, Optimization and Nonstandard Analysis (1994) 185. J. L. Bueso et a/., Compatibility, Stability, and Sheaves (1995) 186. A. N. Michel and K. Wang, Qualitative Theory of Dynamical Systems (1995) 187. M. R. Dame/, Theory of Lattice-Ordered Groups (1995) 188. Z. Naniewicz and P. D. Panagioropoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) 189. L. ./. Corwin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (I 995) 190. L. H. Erbe er a/., Oscillation Theory for Functional Differential Equations (1995) 191. S. Agaian et a/., Binary Polynomial Transforms and Nonlinear Digital Filters (1995) 192. M. /. Gil’, Norm Estimations for Operation-Valued Functions and Applications (1995) 193. P. A. Griller, Semigroups: An Introduction to the Structure Theory (1995) 194. S. Kichenassamy, Nonlinear Wave Equations (1996) 195. V. F. Krorov, Global Methods in Optimal Control Theory (I 996) 196. K. /. Beidar er a/. , Rings with Generalized Identities (1996) 197. V. 1. Arnaurov et a/, Introduction to the Theory of Topological Rings and Modules (1996) 198. G. Sierksma, Linear and Integer Programming (1996) 199. R. Lasser, Introduction to Fourier Series (1996) 200. V. Sima, Algorithms for Linear-Quadratic Optimization (1996) 201. D. Redmond, Number Theory (1996) 202. ./. K. Beem et a/., Global Lorentzian Geometry: Second Edition (1996) 203. M. Fonrana et a/., Prijfer Domains (1997) 204. H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997) 205. C. Q. Zhang, Integer Flows and Cycle Covers of Graphs (I 997) 206. E. Spiegel and C. J. O’Donnell. Incidence Algebras (1997) 207. 8. Jakubczyk and W. Respondek, Geometry of Feedback and Optimal Control (I 998) 208. T. W. Haynes et a/., Fundamentals of Domination in Graphs (1998) 209 T. W. Haynes er a/., Domination in Graphs: Advanced Topics (1998) 210. L. A. D’Alorro et a/., A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998) 21 1. F. Halrer-Koch, Ideal Systems (1998) 212. N. K. Govil et a/., Approximation Theory (I 998) 213. R. Cross, Multivalued Linear Operators (1998) 214. A. A. Marrynyuk, Stability by Liapunov’s Matrix Function Method with Applications (I 998)

215. 216. 217.

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces (1999) A. Manes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances (1999) G. Kafo and D. Struppa, Fundamentals of Algebraic Microlocal Analysis (1999) Additional

Volumes

in Preparation

This Page Intentionally Left Blank

HYPERSPACES Fundamentals

Alejandro

and Recent Advances

Wanes

Universidad National Ciudad Universitaria,

Autdnoma Mkxico

de Mbxico

Sam B. Nadler, Jr. West Virginia University Morgan town, West Virginia

MARCEL

DEKKER, INC.

NEW YORK - BASEL

ISBN: O-8247-1982-4 This book is printed on acid-free paper. Headquarters Marcel Dekker, Inc. 270 Madison Avenue, New York, NY 10016 tel: 2 12-696-9000; fax: 2 12-685-4540 Eastern HemisphereDistribution Marcel Dekker AG Hutgasse 4, Postfach 8 12, CH-400 I Basel, Switzerland tel: 44-6 I-26 I-8482; fax: 44-6 l-26 l-8896 World Wide Web http://www.dekker.com The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the headquarters address above. Copyright0

1999 by Marcel Dekker, Inc. All Rights Reserved.

Neither this book nor any part may be reproduced or transmitted in any form or by any means. electronic or mechanical. including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit) 10987654321 PRINTED

IN THE UNITED

STATES

OF AMERICA

To Margarita

and Elsa

This Page Intentionally Left Blank

Preface We wrote the book with two purposes in mind: to present the fundamentals of hyperspaces in a pedagogically appropriate way and to survey the research aspects of the field. The first book on hyperspaces was written in 1978 and is referred to here as HS (Hyperspaces of Sets, by Sam B. Nadler, Jr., Marcel Dekker, Inc.). HS was written more as a research monograph than as an introductory text. In contrast, we have written this book as a text, as well as a reference. The book emphasizes the hyperspaces 2” and C(X), where X is a continuum. We also include material about symmetric products, containment hyperspaces, selections, spaces of segments and spaces of Whitney levels. We define some concepts differently than in the literature so as to obtain the most appropriate formulations for theorems (e.g., the definition in 13.1 and the theorem in 17.7 compared with the original version of the theorem in HS, p. 79). The book contains historical comments and references to original sources, either incorporated into the main body of the text or at the ends of sections. For a brief history of hyperspaces up until 1978, see HS (pp. xii-xiv). The first six chapters are solely concerned with the fundamentals of hyperspaces, providing a basic overview of the subject and a foundation for further study. The remaining chapters include more specialized material. The book is replete with exercises that we hope will enable the reader to gain a deeper understanding of material and more facility with techniques. A number of exercises contain new ideas, interesting sidelights, applications, and comments; the exercises are therefore an integral part of the book. Examples abound throughout the book. Moreover, we devote all of Chapter II to basic examples; the detailed verifications of the properties of the examples in Chapter II put the techniques used on a solid basis. The book is reasonably self-contained. We provide proofs for almost all results in the first six chapters and for many results in the remaining chapters. When we do not prove a result, we sometimes discuss the ideas V

vi

PREFACE

involved in the proof; we always refer to the literature where a proof can be found. Furthermore, we include basic material (usually with proofs) that is not specifically about hyperspaces: absolute retracts and Z-sets (section 9)) Peano continua (section lo), boundary bumping (section 12), and general theorems concerning the fixed point property (section 21). Of course, we limit our treatment of these topics to ideas and results used in hyperspaces. A tremendous amount of research has been done on hyperspaces in the past twenty years. New developments have begun, and many of the almost two hundred research questions in HS have been partially or completely answered. Therefore, we felt it was time to survey the research in hyperspaces that has occurred since HS appeared. We do this beginning with Chapter VII, covering recent research thoroughly. For example, we include complete details for the especially important solution to the dimension problem (section 731, the characterizations of Class(W) (section 671, the n-od problem (section 70), and the product problem (section 79). The most active research topic in hyperspaces has been Whitney properties and Whitney-reversible properties. We cover the results in Chapter VIII. The table at the end of Chapter VIII gives the reader a bird’s-eye view of the resultsand their relations to eachother. or Problem refers to Starting with Chapter VII, the term Question a question or problem whose answer is not known (at least by us). The last chapter is wholly concerned with questions. The first two sections of the last chapter quote the questions from HS and discuss their current status. The final section discussesmore questions, a number of which are original with this book. After the references for the last chapter is a list of the papers concerned with hyperspaces that have appeared since 1978 (we apologize to those whom we overlooked in our search of the literature, and we ask them to let us know so that we can include their papers in a subsequentprinting of the book). Our general notation is standard. Typographical considerations lead us to use X as well as cl(A) (or cl-~(A)) to denote the closure of A; cl is usually used when the expression over which we are taking the closure consists of several letters and/or symbols (however, we do not mix the two ways of denoting closure in the sameproof, definition, etc.). Other notation is explained as it comes up. We express our gratitude to several people for their help: Janusz J. Charatonik and Sergio Ma&s, who read the manuscript and gave us many beneficial suggestions; Gerard0 Acosta and Fernando Orozco, who helped us search the literature; Joann Mayhew, who patiently and diligently did a splendid job typing a large portion of the book; the people at Marcel Dekker, Inc., especially Maria Allegra, who were always patient and helpful; and the students Daniel Arkvalo, Fklix Capulin, Benjamin Es-

PREFACE

vii

pinoza, Fanny Jasso, Maria de J. L6pez, Jorge Martinez, Ver6nica Martinez de la Vega, Albert0 C. Mercado, Fernando Orozco, Patricia Pellicer and Likin C. Sim6n, who took a one year course based on the manuscript for the book. We also thank the Instituto de Matematicas of the Universidad National Autbnoma de Mexico, and the Mathematics Department of West Virginia University, for the use of resources during the preparation of the book. We express our gratitude to Gabriela SanginCs for formatting the final version of the book. It is our hope that the book will serve well as a text to attract people to the field and that researchers will find the book valuable.

Alejandro Illanes Sam B. Nadler, Jr.

This Page Intentionally Left Blank

Contents Preface

V

Part

One

I.

The

Topology

for

3

Hyperspaces

1. The General Notion of a Hyperspace ..............

3

Topological Invariance ...................... Specified Hyperspaces ......................

5 6

Exercises ............................. 2. The Hausdorff Metric H,.J .................... Proof That Hd Is a Metric ....................

7

A Result about Metrizability

9 11 12

of CL(X) .............

Exercises ............................. 3. Metrizability of Hyperspaces .................. Metrizability of 2x ........................ Metrizability and Compactness of CL(X) Exercises .............................

14 16 16 ...........

18 19

4. Convergence in Hyperspaces ................... L-convergence, TV-convergence ................. Relationships between L-convergence and Tv-convergenece When X Is Compact Hausdorff ................. Countable Compactness Is Necessary .............. Exercises ............................. References ............................... ix

20 20 .

22 25 26 26 28

X II.

TABLE OF CONTENTS Examples:

Geometric

Models

for

Hyperspaces

5. C(X) for Certain Finite Graphs X ............... XanArc ............................. X a Simple Closed Curve .................... XaNoose ............................. X a Simple n-od ......................... Historical Comments ....................... Exercises ............................. 6. C(X) When X Is the Hairy Point ................ Exercises ............................. 7. C(X) When X Is the Circle-with-a-Spiral. ........... Cones, Geometric Cones ..................... The Model for C(X) ....................... Knaster’s Question ........................ When C(Y) x Cone(Y) ..................... Exercises ............................. 8. 2x When X Is Any Countably Infinite Compacturn ...... Cantor Sets ............................ Preliminary Results ....................... Structure Theorem ........................ Uniqueness of Compactifications ................ The Model for 2’ ........................ Exercises ............................. References ............................... III.

2’

and

C(X)

for

Peano

Continua

X

9. Preliminaries: Absolute Retracts, Z-sets, Torunczyk’s Theorem Exercises ............................. 10. Preliminaries: General Results about Peano Continua .... Exercises ............................. 11. The Curtis-Schori Theorem for 2x and C(X) ......... When 2; and CK(X) Are Z-sets ................ The Curtis-Schori Theorem ................... Further Uses of Torunczyk’s Theorem ............. Exercises ............................. References ...............................

31

33 33 35 36 39 44 44 46 50 51 51 53 59 59 62 64 65 65 67 67 70 72 72 75

76 79 80 83 85 85 89 90 91 94

TABLE OF CONTENTS IV.

Arcs

xi

in Hyperspaces

97

12. Preliminaries: Separation, Quasicomponents, Boundary Bumping ............................

97

Exercises .............................

103

13. A Brief Introduction to Whitney Maps

............

105

..................

Definition of a Whitney Map

105

Existence of Whitney Maps ...................

106

Exercises .............................

108

14. Order Arcs and Arcwise Connectedness of 2x and C(X) Definition of Order Arc

.....................

Arcwise Connectednessof 2x and C(X)

...

110 110

............

110

Application: 2’ > I”O ......................

114

Original Sources .........................

116

Exercises .............................

117

15. Existence of an Order Arc from As to Ai

...........

119

Necessary and Sufficient Condition ...............

119

Application: Homogeneous Hyperspaces ............

122

Original Sources .........................

124

Exercises .............................

124

16. Kelley’s Segments ........................

127

Kelley’s Notion of a Segment ..................

127

Results about Segments .....................

128

Addendum: Extending Whitney Maps .............

131

Original Sources .........................

132

Exercises .............................

132

17. Spacesof Segments, S,(E)

...................

134

Compactness ........................... S,(R) M 8(B) S,(Z’),S,(C(X)) Application:

134

.......................... When X Is a Peano Continuum

136 ......

Mapping the Cantor Fan Onto 2x and C(X)

138 . . 140

Original Sources .........................

141

Exercises .............................

141

TABLE OF CONTENTS

xii

18. When C(X) Is Uniquely

Connected

. . . .

143 145

Uniqueness of Arcs in C(X) When X Is Hereditarily Indecomposable . . . . . . . . . . . . . . . . . . .

146

Original Exercises References Shape

Sources

Theorem

. . . . . . . . . . .

. .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

148 148

. .

148

. . . . . . . . . . . . . . . . . . . . . . . .

149

and Contractibility

153

19. 2’ and C(X)

of Hyperspaces

as Nested Intersections

of ARs

.........

153

2’, C(X) Are Acyclic

......................

155

2’, C(X) Are crANR.

......................

155

2’, C(X)

Are Unicoherent

....................

Whitney

Levels in C(X)

2’, C(X)

Have Trivial Sources .........................

Original Exercises

157 .............

Are Continua Shape ..................

.............................

.................... 20. Contractible Hyperspaces The Fundamental Theorem ...................

159 160 161 161 164 164

X Contractible, X Hereditarily Indecomposable ........ Property (K) (Kelley’s Property) ................

166

Theorem about Property (K) .................. X Peano, X Homogeneous ....................

168

Original Sources ......................... Exercises .............................

175

References ............................... VI.

. .

Structure of Arcs in C(X) When X Is Hereditarily Indecomposable . . . . . . . . . . . . . . . . . . .

The Characterization

V.

Arcwise

Hyperspaces

and the Fixed Point Property

167 173 176 177

181

21. Preliminaries: Brouwer’s Theorem, Universal Maps, Lokuciewski’s Theorem . . . _ . . . . . . . . . . . . . . . . . 181 Original Sources . . . . . . . . _ . . . . . . . . . . . . . . . . 186 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

.. . x111

TABLE OF CONTENTS 22. Hyperspaces with the Fixed Point Property .......... Peano Continua. ......................... Arc-like Continua. ........................ Circle-like Continua ....................... A General Theorem ....................... Dendroids ............................. Hereditarily Indecomposable Continua ............. Addendum: Dim[C(X)] > 2 ................... Original Sources ......................... Exercises ............................. References ...............................

Part VII.

187 187 190 192 193 196 196 197 197 199

Two 205

Whitney Maps 23. Existence and Extensions .................... Exercises ............................. 24. Open and Monotone Whitney Maps for 2’ Exercises ............................. 25. Admissible Whitney Maps ................... Exercises .............................

..........

26. A Metric on Hyperspaces Defined by Whitney Maps ..... Exercises ............................. References ............................... VIII.

187

Whitney Properties and Whitney-Reversible Properties 27. Definitions ............................ Exercises ............................. 28.ANR ............................... Exercises ............................. 29. Aposyndesis ........................... Exercises ............................. 30. AR ................................ Exercises .............................

205 206 207 215 216 225 227 227 228

231 231 233 234 236 238 239 239 245

xiv

TABLE OF CONTENTS 245 246 247 247

31. Being an Arc ........................... ............................. Exercises ......................... 32. Arc-Smoothness ..................... 33. Arcwise Connectedness ............................. Exercises .......................... 34. Being Atriodic

Exercises

.............................

35. C*-Smoothness, Class(W) and Covering Property ............................. Exercises ............. 36. Tech Cohomology Groups, Acyclicity ................... 37. Chainability (Arc-Likeness) ............................. Exercises 38. Being a Circle .......................... ............................. Exercises .......................... 39. Circle-Likeness ............................. Exercises 40. Cone = Hyperspace Property .................. Exercise .............................. .......................... 41. Contractibility ............................. Exercises 42. Convex Metric .......................... ............................. Exercises ............................ 43. Cut Points

............................. Exercises ......................... 44. Decomposability Exercises

.............................

............................ 45. Dimension ............................. Exercises ...................... 46. Fixed Point Property ............................. Exercises 47. Fundamental Group ....................... ............................. Exercises ........................... 48. Homogeneity Exercise .............................. ........................... 49. Irreducibility

Exercises

251

.............................

......

251 253 253 256 257 257 259 259 259 259 260 261 262 262 264 265 265 265

267 267

268 268 269 270 270 271 271 271 272 273

276

xv

TABLE OF CONTENTS 50. Kelley’s Property ......................... ............................. Exercises ......................... 51. X-Connectedness

276 278 279 280

............................. 52. Local Connectedness ....................... ............................. Exercises ......................... 53. n-Connectedness ............................. Exercises Exercises

54. Planarity Exercises

281 281 281 283

............................. .............................

............................ ............................. ............................ 56. Pseudo-Arc .............................. Exercise 57. Pseudo-Solenoids and the Pseudo-Circle ........................... 58. R3-Continua Exercise ..............................

283 284 284

55. P-Likeness Exercises

285 286 286 286 286

............

287

59. Rational

Continua ........................ ............................. Exercises 60. Shape of Continua ........................ 61. Solenoids ............................. 62. Span. ............................... ........................... 63. Tree-Likeness

287 287 287 290

64. Unicoherence ........................... ............................. Exercises

292 293

Table Summarizing Chapter References ............................... IX.

VIII

291 292

................

Whitney Levels .......................... 65. Finite Graphs Exercises ............................. 66. Spaces of the Form &(X, t) Are ARs Exercises 67. Absolutely Exercises

............................. C*-Smooth, Class(W) .............................

294 299 305 305 313 314

.............

and Covering

Property

318 . . 319 325

xvi

TABLE OF CONTENTS 68. Holes in Whitney Levels ............... References .........................

X.

.

XII.

............

347

of C(X)

72. Previous Results about Dimension of Hyperspaces ............................. Exercises 73. Dimension of C(X) for a-Dimensional Continua X ............................. Exercises

......

74. Dimension of C(X) for l-Dimensional References ...............................

......

between

76. Retractions between Hyperspaces ....................... Exercises 77. Induced Maps .................... ....................... Exercises References ......................... XIII.

Continua

X

358 359

. . . . 363 . . . . 368 , . . . 371 . . . . 379

..........

. . 1 . 381 . . . . 387 . . . . 390 395 395 ........

395 399 400

.....................

Spaces of Finite Subsets ........................... Admissibility Hyperspace

......

347 348 349 357

Hyperspaces

More on Contractibility of Hyperspaces ............... 78. More on Contractible Hyperspaces Contractibility vs. Smoothness in Hyperspaces R3-Sets ..............................

Maps Preserving

326

342 344 345

...............................

Special Types of Maps 75. Selections ....................... ....................... Exercises

.

337 341

Exercises ............................. 71. Neighborhoods of X in the Hyperspaces Exercises .............................

Dimension

.

333 333 336

Exercises ............................. 70. Cells in Hyperspaces .......................

XI.

9

. . . . . 329

General Properties of Hyperspaces 69. Semi-Boundaries .........................

References

.

402 Contractibility

..........

403

xvii

TABLE OF CONTENTS More on Kelley’s Property .................... Exercises ............................. References ...............................

405 406 408

XIV. Products, Cones and Hyperspaces ............... 79. Hyperspaces Which Are Products ............................. Wrinkles. Folds. ............................... Proof of the Main Theorem ................... ............................. Exercises 80. More on Hyperspaces and Cones ................ ............................. Exercises References ...............................

413 413 414 415 421 423 424 431 434

XV.

437

Questions 81. Unsolved and Partially Solved Questions of [56] ........ 82. Solved Questions of [56] ..................... 83. More Questions .......................... General Spaces .......................... Geometric Models ........................ Z-Sets ............................... ....................... Symmetric Products Size Maps ............................. The Space of Whitney Levels for 2x .............. Aposyndesis ............................ Universal Maps .......................... References ............................... Literature

Special Index

Related to Hyperspaces

Symbols

of Continua

Since 1978 ....

437 463 470 470 471 471 471 472 473 473 473 474 478 497 499

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HYPERSPACES

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Part

One

This Page Intentionally Left Blank

I.

The Topology Hyperspaces

1.

The General

Notion

for

of a Hyperspace

Let X be a topological space with topology T. A hyperspace of X is a specified collection of subsets of X with the Vietoris topology (which we will define in 1.1). For convenience, we exclude the empty set 0 from being a point of a hyperspace. Also, to avoid pathology, we restrict our attention to hyperspaces whose points are closed subsets of X (cf. Exercise 1.13). Thus, the largest hyperspace of X is CL(X)

= {A c X : A is nonempty and closed in X}

with the Vietoris topology, which we now define. 1.1 Definition. Let (X, T) be a topological space. The Vietoris topology for CL(X) is the smallest topology, TV, for CL(X) having the following natural property: (-4 E CL(X) : A C U} E TV whenever U E T, and {A E CL(X) : A c B} is TV-closed whenever B is T-closed. We may exhibit a base, Bv, for TV as follows. First, note the following general notation: For any finitely many subsets Sr , . . . , S, of X, let (Sl,...,

S,) = {A E CL(X)

1.2 Theorem.

: A c lJ~==,Si and A II Si # 0 for each i}.

Let (X, T) be a topological space, and let

Bv = {(VI,...

, U,,) : Vi E T for each i and n < oo}.

Then Bv is a base for TV. 3

4

I. THE TOPOLOGY FOR HYPERSPACES

Proof. The sets that are used to define TV in 1.1 can be expressed using the notation for the members of BV as follows: for any U E T and any T-closed set B, {A E CL(X) : A c U} = (U); {A E CL(X) : A c B} = CL(X) - (X,X - B). Hence, the definition in 1.1 says that TV is the smallest topology for CL(X) containing all the sets of the forms (U) and (X,U) for U E T. In other words, letting S = {(U) : U E T} U {(X,U)

: U E T},

the collection S is a subbase for TV, which means that the family, S*, of all finite intersections of members of S is a base for TV. Hence, it suffices to prove that S* = Bv. To seethat 13~ c S*, let VI, . . . , U,, E T(n < oo) and simply observe that , Un) = (ULW

Wl,...

f-l (q-=1(X,

Vi)).

Finally, we prove that S’ c Bv. First, we prove that if Z.4,W E Bv, then U n W E Bv. To prove this, assume that U = (VI,. . . , Uk) and W = (WI,, . . , W,), where Vi, Wi E T and k, m < 00. Let U = Ut=,Ui and let W = Uz”=, Wi. Then, it is easy to check that U n w = (VI n W, . . . , Uk n W, WI

n U, . . . , W, n U).

Therefore, U n W E Bv. Thus, to prove that S’ c Bv, it suffices to prove that the intersection of any two members of S is a member of Bv. (Note: If & denotes the empty subcollection of S, then, clearly, nE E S*; however, since & is a collection of subsetsof CL(X), we see that nE = CL(X), which implies that n& E BV since CL(X) = (X).) We show that the intersection of any two members of S is a member of BV in (l)-(3) below (where UI, U2 E T): (I)

(u,)

n

(u2)

(2)

(ud

n

Ku2)

=

(uln =

u2); Wdmu2);

(3) (x, W n (x, U2) =

(x,

VI,

u2).

This completes the proof that S’ c Bv.

n

The theorem in 1.2 leads us to think of the Vietoris topology in a geometric way, as in Figure 1 (top of next page), where A and B are points in the same member (VI, . . . , U,,) of the base 0~.

TOPOLOGICAL

A,B E

INVARIANCE

5

(&,...,u5)

Figure 1

Topological

Invariance

We prove the fundamental theorem about the topological CL(X). The symbol M means “is homeomorphic to.” 1.3 Theorem.

If X x Y, then CL(X)

invariance

of

x CL(Y).

Proof. Let h be a homeomorphism from X onto Y. Define a function, h’, on CL(X) as follows: h*(A) = h(A) for each A E CL(X). Since h is a closed map, h’ maps CL(X) into CL(Y); moreover, h’ maps CL(X) onto E CL(X) and h*(h-l(B)) = B. CL(Y) since for any B E CL(Y), h-‘(B) Since h is one-to-one, we seethat h* is one-to-one.

Now, note the two equalities below (the proof of the first one usesonly that h is a closed map; the proof of the second one usesthat h is one-to-one and maps X onto Y and that h is continuous): (1) (h*)-‘((Wr,...,W,)) open in Y; (2) h*(W,...

= (h-‘(W~),...,h-‘(W,)),

where each IV, is

,Um)) = (h(U,), . . . , h(Um)), where each U, is open in X.

By (l), the fact that h is continuous, and 1.2, we seethat h’ is continuous. By (2), the fact that h is an open map, and 1.2, we seethat h” is an open map and, hence, that (h*)-’ is continuous.

I. THE TOPOLOGY

6

FOR HYPERSPACES

Therefore, we have proved that h* is a homeomorphism from CL(X) onto CL(Y). I The converse of 1.3 is false. This is easily seen by letting X and Y be nonhomeomorphic, indiscrete spaces. However, much more satisfying examples come from 8.10 and 11.3 (regarding 8.10, seethe secondparagraph of section 8). We comment about the proof of 1.3. The idea of forming h* from h when h is continuous is an important idea in hyperspace theory. For example, results about the fixed point property for hyperspaces can be proved using this idea (see section 22). For more information, seesection 77 (where h* is denoted by 2h and h’ /C(X) is denoted by C(h)).

Specified

Hyperspaces

By considering simple topological properties - connectedness,compactness, etc. - we are led to consider hyperspaces that consist of sets having one or more of these properties. We introduce some of these hyperspaces in 1.4-1.8. Of course, each of the hyperspaces has the subspace topology obtained from the Vietoris topology on CL(X). = {A E CL(X)

1.4 Definition.

CLC(X)

1.5 Definition.

2x = {A E CL(X)

: A is connected}.

: A is compact}.

We note that 2x = CL(X) when X is compact. We also note that when X is Hausdorff, 2x = {A c X : A is nonempty and compact}. 1.6 Definition.

C(X)

= {A E 2dY : A is connected}; in other words,

C(X) = 2x n CLC(X). Starting with Chapter II, the book is almost exclusively about the two hyperspaces 2x and C(X) when X is a compact metric space. For the next definition, (Al denotes the cardinality of a set A. Also, recall that a Tl-space is a topological space, X, such that (2) is closed in A’ for each 2 E X. 1.7 Definition. Assume that X is a Ti-space. For each n = 1,2,. . . , let F,,(X) = {A c X : 1 5 IAJ 5 n}, which we consider as a subspace of CL(X) or 2”. The space F,(X) is called the n-fold symmetric product of X; the l-fold symmetric product of X is also called the space of singletons (note Exercise 1.15).

Symmetric products provide a simple way to obtain interesting spaces. In particular, the n-fold symmetric product of X and the Cartesian product

7

EXERCISES

of n copies of X are usually quite different. This is true even when X is a simple metric continuum and n = 2. For example, see Exercise 1.26. Basic open questions about symmetric products are in 83.11-83.14; also, see 78.19 and 78.20. 1.8 Definition. For a Ti-space, X, let F(X) F(X) the space of finite subsetsof X.

= U~i,,F,(X).

We call

An open problem and related results about F(X) are in 78.19 and the comment following 78.20. We conclude this section by noting that the topological invariance of CL(X) in 1.3 is also true of each of the hyperspaces in 1.4-1.8: 1.9 Theorem. If X = Y, then ‘tli(X) z ‘&(Y), where ‘&(.X) and xi(Y) are the hyperspaces of X and Y defined in 1.i for each i = 4,. . . ,8 (we assumethat X and Y are Ti-spaces when i = 7 or 8). Proof. For each i, the homeomorphism h’ in the proof of 1.3 maps the hyperspace ?&(X) onto the hyperspace ?&(Y). n

Exercises 1.10 Exercise. Let (X,T) be the Sierpinski two-point space: X = (0, 1) with the topology T = {X,0, (0)). What familiar space is (C-W), TV)? Describe the space (CL(X), TV) when X = (0, 1,2} with the topology T consisting of the sets X, 0, (0, I}, { 1,2}, and { 1). 1.11 Exercise. For any topological space X, CL(X) is a To-space. (Recall that a To-space is a topological space, (Y,T), such that for any ~1, ~2 E Y with yi # ~2, there exists U E T such that yi E U and ?/j $ U for some choice of i and j.) 1.12 Exercise. converse is false. 1.13 Exercise.

If X is a Ti-space, then CL(X)

is a Ti-space. The

Let (X, T) be a topological space, and let N(X)

= {A c X : A #

0).

The Vietoris topology for CL(X) extends naturally to what we would call the Vietoris topology for N(X): simply replace CL(X) with N(X) throughout 1.1. However, we did not consider the hyperspace N(X) because it almost always fails to have decent separation properties. In fact, prove that

8

I. THE

N(X) with its Vietoris space.

TOPOLOGY

FOR

HYPERSPACES

topology is not a Ti-space unless (X, T) is a discrete

1.14 Exercise. If X is a regular space, then CL(X) is a Hausdorff space. If CL(X) is a Hausdorff space and X is a Ti-space, then X is a regular space. 1.15 Exercise.

If X is a Tr-space, then X M Fi(X).

1.16 Exercise. If X is a Hausdorff space, then F,(X) is closed in CL(X) for each n = 1,2,. . .. Give an example of a Ti-space, X, such that Fl (X) is not closed in CL(X). 1.17 Exercise. If X is a connected Tr-space, then F,(X) for each n; hence, CL(X) is connected.

is connected

1.18 Exercise. If X is a Ti-space, then X is separable if and only if CL(X) is separable. (Note: the proof of the “if” part usesthe Axiom of Choice.) 1.19 Exercise. X. Let

Let X be a Ti-space, and let K be a closed subset of CLK(X)

Then, CLK(X)

= {A E CL(X)

: A > K}.

is closed in CL(X).

1.20 Exercise. CL(X). 1.21 Exercise.

If X is a normal space, then CLC(X)

is closed in

Let (X, T) be a Ti-space. For any Ui , . . . , lJ,, E T (n
0 and any A E CL(X),

let

: a E A}.

let

Nd(r, A) = {x E X : d(z, A) < T}

(Figure 2). We call Nd(r, A) the generalized open d-ball radius

in X about A of

r.

Note that (a) and (b) above say that A c Nd(r,B) and B c Nd(r,A), respectively. Hence, the following definition is simply a reformulation of what we referred to as the “Vietoris distance” from A to B.

Nd(r, A) Figure 2

PROOF

THAT

Hd Is A METRIC

11

2.1 Definition. Let (X, d) be a bounded metric space. The Huusdorff metric for CL(X) induced by d, which is denoted by Hd, is defined as follows: for any A,B E CL(X), Hd(A, B) = inf {r > 0 : A c Nd(r, B) and B c Nd(r, A)}.

Proof

That

Hd Is a Metric

We prove that Hd is, indeed, a metric. 2.2 Theorem.

If (X, d) is a bounded metric space,then Hd is a metric.

Proof. We begin by noting that, since d is a bounded metric, Hd is a real-valued (nonnegative) function. We next observe, as is evident from 2.1, that Hd is a symmetric function; in other words, Hd(A, B) = Hd(B, A) for all A, B E CL(X). Now, assume that A,B E CL(X) such that Hd(A,B) = 0. Then, by 2.1, A c N~(E,B) for all E > 0. Hence, fixing p E A, there exists b, E B for each n = 1,2,. . . such that d(p,M < l/n. Thus, since {b,}~zl converges to p and B is closed in X, p E B. Therefore, we have proved that A c B. A similar argument shows that B c A. Hence, A = B. The converse - Hd(A, A) = 0 for all A E CL(X) - follows immediately from 2.1. It only remains to prove the triangle inequality for Hd. For this purpose, it is useful to have the following fact (which is an elementary consequence of definitions): (#) for any K, L E CL(X) Now, let A, B, C E CL(X). &(A,

and E > 0, K C Nd(Hd(K, L) + E, L).

We prove that c) 5 &(A,

B) + Hd(B, c).

Let E > 0. Let a E A. Then, by (#), there exists b E B such that (1) d(a, b) < Hd(A, B) + E. Since b E B, we seeby using (#) again that there exists c E C such that (2) d(b, c) < Hd(B, C) + E. By (l), (2), and the triangle inequality for d, we have that (3) d(a, c) < Hd(A, B) + Hd(B, C) + 2~.

12

I. THE TOPOLOGY

FOR HYPERSPACES

Thus, since a was an arbitrary point of A, we have proved that (4) A C Nd(&(A,

B) + H@,

c) + 25, c).

A similar argument (which starts with a point of C) shows that (5) c c Nd(Hd(c, B) + Hd(B, A) + 2~, A). Recall from the second paragraph of the proof that Hd is a symmetric function; hence, we may rewrite (5) as follows: (6) c C Nd(Hd(A, B) f Hd(B, c) + 2E, A). By (4), (6), and 2.1, Hd(A,C) 5 Hd(A,B) since E > 0 was arbitrary, Hd(A, C) 5 &(A,

A Result

about

+ Hd(B,C) + 2s. Therefore, B) + Hd(B, C). n

Metrizability

of CL(X)

The discussion at the beginning of the section shows how the idea for the Hausdorff metric arises naturally from examining the basic open sets (VI,... , Un) for the Vietoris topology. In particular, when the metric d for X is bounded, the conditions in (a) and (b) at the beginning of the section seem to describe the interpretation of the Vietoris topology in Figure 1, p. 5. Thus, when X has a bounded metric, we would expect that the Vietoris topology for CL(X) and the Hausdorff metric topology for CL(X) would be the same. However, this is not so. In fact, for Ti-spaces X, the Vietoris topology for CL(X) is not even metrizable (by any metric) unless X is compact. The following lemma gives the principal reason (as you will seefrom the proof of the succeeding theorem). 2.3 Lemma. If Y is an infinite, discrete space, then the Vietoris topology for CL(Y) does not have a countable base. Let ,0 be a base for the Vietoris topology for CL(Y). Note that Proof. each A E CL(Y) is open in Y. Hence, for each A E CL(Y), there exists a,4 E /I such that A E f?A C (A). Thus, it follows easily that (1) A = USA for each A E CL(Y). Now, it follows immediately from (1) that if A, A’ E CL(Y) such that A # A’, then DA # a,&. In other words, the function A + f?jA is a one-toone function from CL(Y) into p. Hence,

A RESULT ABOUT METRIZABILITY c-4

ICW)l

OF CL(X)

13

I IPI-

Since Y is a discrete space, CL(Y) = {A c Y : A # 0}; thus, since Y is an infinite set, CL(Y) is uncountable. Therefore, by (2), the base p is uncountable. w 2.4 Theorem. Let (X, T) be a Ti-space. then (X, T) is a compact, metrizable space.

If (CL(X),

TV) is metrizable,

Proof. Since (CL(X),Tv) is metrizable and X M Fi(X) (by 1.15), it is clear that (X,T) must be metrizable. We prove that (X,T) is compact. Suppose that (X, T) is not compact. Then, since (X, T) is metrizable, there is a countably infinite, closed (in X), and discrete subspace, (Y, TIY), of (X,T). Since Y is closed in X, we have by 1.22 that CL(Y) with its own Vietoris topology (TIY) v is a subspace of (CL(X), TV). Thus, since we are assuming that (CL(X), 2’“) is metrizable, we have that (1) (CL(Y),

(TIY)“)

is metrizable.

Now, note that (Y, TIY) is a Z’i-space (since (Y, T)Y) is discrete); also, note that (Y,TIY) is separable (since Y is countable). Hence, by 1.18, (2) (CL(Y),

(TIY)v)

is separable.

By (1) and (2), there is a countable base for (TIY)“. However, infinite, discrete space, this contradicts 2.3. Therefore, (X,T)

Y being an is compact.

n

The theorem in 2.4 is somewhat disappointing since it limits the generality in which (CL(X),T v ) is metrizable to compact spaces X. However, we will be consoled by the important theorems in the next section: They show that for any metric space (X, d), the hyperspace 2x with its Vietoris topology is metrizable by the Hausdorff metric; they also show that the converse of 2.4 is true and, moreover, that it is the Hausdorff metric that does the metrization. 2.5 Remark. Let (X, d) be a metric space. Let A, B E 2x (i.e., A and B are nonempty, compact subsets of X). Then there is a real number T > 0 such that A C Nd(r, B) and B C Nd(r, A) (e.g., r = l+ diameterd(AUB)). Thus, whether d is bounded or not, the formula for Hd(A, B) in 2.1 produces a real number when A, B E 2x. Now, recall that the assumption in 2.2 that d is bounded was only used at the beginning of the proof of 2.2 to know that Hd is real valued. Therefore, the proof of 2.2 shows that Hd is a metric for 2x whether d is bounded or not. We call this metric the Hausdorff metric

14

I. THE TOPOLOGY

FOR HYPERSPACES

for 2 x induced by d. We use the same symbol, Hd, for this metric that we

used in 2.1 (since we know that doing so will never cause any confusion).

Exercises 2.6 Exercise. Let (X, d) be a bounded metric space. Let A, B, A’, B’ E CL(X) such that A’ C A and B’ c B. Then &(A

2.7 Exercise.

U B’, B U A’) 5 Hd(A, B)

Let (X,d)

be a bounded metric space. Then, for any

A, B E CL(X),

Hd(A, B) = ma ‘1:~ d(a,B), ;F; d(b,A)}. 2.8 Exercise. Let (X, d) be a metric space. If A, B E 2x, then there are points, p E A and q E B, such that Hd(A, B) = d(p, q). Also, when (X, d) is a bounded metric space, is the analogous result for A, B E CL(X) true? 2.9 Exercise. Let (X, d) be a metric space. If A, B E 2” and if A c Nd(r, B) and B C Nd(r, A), then &(A, B) < r. Is the analogous result for A, B E CL(X) true (assuming, of course, that (X, d) is a bounded metric space)? 2.10 Exercise. Use modifications of the proof of 2.4 to prove the following result. Let (X,T) be a Tr-space. If (CL(X),Tv) has a countable base, then (X, T) is compact. Can you also conclude, as in 2.4, that (X,5”) is metrizable? 2.11 Exercise. Even though Hd # Hdt whenever d # d’, there is a much more important reason to involve the given metric on X in our notation for the Hausdorff metric for CL(X): two bounded metrics, d and d’, may induce the sametopology on X, yet the Hausdorff metrics Hd and ff& may induce nonhomeomorphic topologies on CL(X). Verify that this can happen by letting X = { 1,2,. . .} and by using the metrics d and d’ defined as follows: 1, ifz#y d(s,y) = 0, ifz=y. { and d’(s,y)=/i-iIforallx,yEX.

EXERCISES

15

2.12 Exercise. If (X, d) is a totally bounded metric space, then THE C TV, where THY is the topology for CL(X) induced by Hd. (A metric space, (X, d), is said to be totally bounded provided that for each E > 0, X is the union of finitely many sets each of which has diameter less than E.) 2.13 Exercise. Let (X,d) be a bounded metric space. Then (l), (2), and (3) are equivalent: (1) (X,d) is totally bounded; (2) (CL(X), Hd) is totally bounded; (3) (CL(X), Hd) is separable. Note that the equivalence of (1) and (3) makes 2.11 trivial. 2.14 Exercise. Let (X, d) be a bounded metric space. If (X,d) is complete, then (CL(X), H d ) is complete. (A metric space is said to be complete provided that every Cauchy sequence with respect to the given metric converges.) For each n = [Hint: Let {Ai}zl b e a Cauchy sequence in CL(X). 1,2,. . ., let Y, = d(UgO,,Ai). Consider Y = IIF&Y,.] Remark.

An important

consequence of Exercises

2.13 and 2.14 is in

3.5. 2.15 Exercise. If (X,d) is a complete metric space, then (2”,Hd) is complete. [Hint: Use Y in the hint for 2.14. Here, A, E 2x for each i; prove that E;, hence each Y,, is totally bounded and complete, therefore compact [5, P. WI 2.16 (i.e., d(s, for Fz(I) First, R2. Next, that

Exercise. Let I = [0, 11, and let d denote the usual metric for I t) = Js - tl for all s, t E I). Let Hd denote the Hausdorff metric induced by d. d ) is topologically embeddable in the plane prove that (Fz(l),H find A~,Az,BI,Bz Hd(Ai,

E Fz(I),

Bj) = iHd(Al,

with

AI # A:! and BI # BP, such

AZ) whenever

i # j.

Therefore, even though (Fz(I), Hd) is topologically embeddable in R2, we see that (Fz(l),H d ) is not isometrically embeddable in any Euclidean space R* (with the metric given by JC~=“=,(G - yiJ2 for (G)L, (yi)L E R”). (An isometry is a distance preserving map between two metric spaces.)

16

3.

I. THE TOPOLOGY

FOR HYPERSPACES

Metrizability

of Hyperspaces

We determine when (CL(X),Tv) and (2x, T~l2~) are metrizable. We show that when they are metrizable, they are metrizable by the Hausdorff metric. We first set down some convenient notation. We will be concerned with 2x and (later on) with hyperspaces, 31, contained in 2*. We could expand on previous notation to fit this situation as follows: (U,, . . . ,Un) nX, HdlR x ‘H. TV\‘% However, the expanded notation is cumbersome. Thus, we often simply use the previous notation and rely on the context for clarification. We also rely on the context for clarification regarding the following notation. Let (X,d) be a metric space, and let 3t c 2x. Then, THY denotes the topology for % induced by the Hausdorff metric Hd for ?t (cf. 2.5). Also, for any A E ?l and T > 0, BH, (T, A) denotes the open Hd-ball in 31 with radius r and center A; in other words, aH,(T,A)

= {B E ‘?i : Hd(A,B)

Metrizability

TV. Taking into account our agreement about notation (above), we seefrom section 1 that TV is the topology for 2x generated by the sets (17) and (X, U) for all U E T (where T is the topology on X). Therefore, to prove that THY > TV, it suffices to show that (U) E THY and (X, U) E THY for all U E T. SO, let U E T. We show that (U) E THY as follows. Note that if U = X, then (U) = 2* and, hence, (U) E THY. Therefore, we assume that U # X. NOW, let A E (U). Let E = d(A,X

- U)(= inf{d(a,z)

: a E A and z E X - U}).

Note that E > 0 since A is a compact subset of the open set U (E exists since A and X - U are nonempty). Furthermore, we seethat BHd(&, A) C (U) as follows: if B E BHd(E, A), then Hd(A, B) < & and, thus, it follows from 2.1 that B C Nd(&,

4;

METRIZABILITY

OF 2’

17

hence, by the way E was defined, we see that B c U, i.e., B E (U). Thus, starting with A E (U), we showed that a,(~, A) C (U) for some E > 0. Therefore, we have proved that (1)

(u)

ETH~.

We show that (X, U) E THY as follows. Let A E (X, U). Then, Afl U # 8. Let p E An U. Since p E U and U E T, there exists 6 > 0 such that

We prove that BH~(~,A) 6. Thus, by 2.1,

C (X,U).

Let B E BHd(6,A).

Then, &(A,B)


0. Therefore, we have proved that

Since we have proved (1) and (2) for any U E T, it follows (as noted above) that THY > TV. We now prove that THY c TV, By 1.2, it suffices to show that for each open Hd-ball, 23~~(T, A), there are finitely many, open subsets, UI, . . . , U,,, of X such that

AE (VI,...

,un)

C aH,(T,A).

To prove this, let A E 2x and let r > 0. Then, since A is compact and nonempty, there are finitely many, open subsets, Ui , . . , , U,, of X satisfying (3)-(5) below: (3) A c u&Vi; (4) A II Vi # 0 for each i; (5) diameter (Ut) < T for each i. It is evident from (3) and (4) that A E (VI,. . . , Un). Therefore, we will be done with the proof once we show that

WI,...

,un)

C BH~(~,A).

I. THE TOPOLOGY

18

FOR HYPERSPACES

To show this, let K E (Vi,. . . , Un). Then, since from (4) and (5) that K c Nc+(r, A);

K c

Uy==, U,, it follows

also, since K fl Vi # 0 for each i, it follows from (3) and (5) that

A c N~(T, K). Hence, Hd(A, K) < r by Exercise 2.9; thus, K E f?~,(r, Therefore, we have proved that THY c TV. n

A).

Let us note that we can state 3.1 in the following way. 3.2 Theorem. If (X, T) is a metrizable topological space, then (2x, TV) is metrizable; moreover, if d is any metric for X that induces T, then

T\J = THY. The converse of the first part of 3.2 is true for Ti-spaces: 3.3 Theorem. Let (X, T) be a Tr-space. Then, (2”) if and only if (X,7’) is metrizable.

T~I) is metrizable

Proof. Assume that (2x, TV) is metrizable; then, since Fi(X) c 2~~ and X z Fi (X) (by Exercise 1.15), we seethat (X, T) is metrizable. The converse is in 3.2. n

Metrizability

and Compactness

of CL(X)

Compare the result about 2x in 3.3 with the following result about

CL(X). 3.4 Theorem. Let (X, T) be a Ti-space. Then, (CL(X), able if and only if (X, T) is a compact, metrizable space.

TV) is metriz-

Proof. Assume that (X,T) is a compact, metrizable space. Then, is (2”, TI/) is metrizable by 3.2 and 2x = CL(X). Therefore, (CL(X),Tv) metrizable. The converse is in 2.4. n The next theorem is fundamental to the development of the theory of hyperspaces. The most general version of the theorem is in Exercise 3.12. 3.5 Theorem. If (CL(X),Tv) is compact.

(X,T)

is a compact, metrizable space, then

EXERCISES

19

Proof. Let d be a metric for X that induces T. Note that (X,d) is totally bounded and complete. Hence, (CL(X), Hd) is totally bounded and complete by Exercises 2.13 and 2.14. Thus, (CL(X), Wd) is compact [5, p. 201. Now, note that CL(X) = 2x (since X is compact); hence, by 3.1, THY = TV. Therefore, (CL(X), TV) is compact. n 3.6 Corollary. Let (X, T) be a Ti-space. If (CL(X), then (CL(X), TV) is compact. Proof.

The corollary follows at once from 3.4 and 3.5. n

3.7 Corollary. If (C(X), TV) is compact. Proof.

TV) is metrizable,

(X,T)

is a compact, metrizable space, then

The corollary follows immediately from 3.5 and Exercise 1.20.

n

3.8 Remark. Assume that (X,T) is a metrizable topological space. We will often want to think of the Vietoris topology on 2x in terms of the Hausdorff metric. The theorem in 3.2 says that we may do this without taking into account any specific metric d that induces T. Thus, we will often omit d from our notation; in other words, we will denote the Hausdorff metric for 2-x by H and the associated topology by TH. In particular, when we say that H is the Hausdorff metric for 2-x, we mean that H is the metric Hd obtained from some metric d for X that induces T. Finally, for z c 2”, we conform to our agreement about notation at the beginning of the section; namely, we use H and TH to denote the Hausdorff metric for ‘?f and the topology for 7-i, respectively. Regarding the comments in 3.8, we must mention that, on occasion, a specific metric (or type of metric) for X that induces T is important when considering properties of 2”. On such occasions, we revert to the notation Hd and THY.

Exercises 3.9 Exercise. Let (X,T) be a compact Hausdorff space. Then, (CL(X),T”) is metrizable if and only if B(X) is a G&-set in CL(X). (A subset, 2, of a topological space, (Y, T), is said to be a Ga-set in Y provided that Z = f$ZiUt, where U, E T for each i.) 3.10 Exercise. Give an example of a topological space, (X,T), that (CL(X),Tv) is metrizable but (X,T) is not metrizable.

such

20

I. THE TOPOLOGY

FOR HYPERSPACES

3.11 Exercise. Let (X, T) be a met&able space. Then, (CL(X), is metrizable if and only if (CL(X), TV) is compact.

TV)

3.12 Exercise. We know that the following result is true when (X, T) is metrizable (combine 3.5, 3.11, and 2.4): If (X, T) is a topological space, then (CL(X), TV) is compact if and only if (X,T) is compact. Prove this general result by using the Alexander Subbase Lemma, which we state as follows: Let (X, T) be a topological space, and let S be a subbase for T; if every cover of X by membersof S has a finite subcover, then (X, T) is compact [5, p. 41. [Hint:For the “if” part, assumethat CL(X)=(UiE~(Uz))U(U+y(X, V,)), where Vi, Vj E T for all i and j (recall S in the proof of 1.2); then consider when u~~JV~ = X and when u~~JV~ # X. For the “only if” part, be careful - {z} may not be a point of CL(X) for 5 E X.1

4.

Convergence

in Hyperspaces

We describe convergence of sequencesin CL(X) directly in terms of the topology on X. Our description enables us to picture convergence as if it were taking place in X. By viewing convergence in CL(X) as occurring in X, we will often enhance our understanding of various aspects of the theory of hyperspaces. Besides, it is a natural tendency to want to think of hyperspace phenomena in terms of the base space X as much as possible: the space X seemssimpler, lessformidable, than the hyperspace CL(X). In 4.2 we define a notion of convergence for sequencesof subsets of X; we call the notion L-convergence. We will see that convergence (of sequences)in CL(X) implies L-convergence when X is regular; we will also see that the converse implication is true when X is countably compact. Hence, convergence in CL(X) and L-convergence are equivalent when X is a compact Hausdorff space. These results are sufficient for our purpose since the rest of the book is almost entirely about hyperspaces of compact metric spaces;moreover, the results can not be much more general (as you will see, notabIy by 4.9).

L-convergence,

TV-convergence

We will define L-convergence using the two companion notions in the following definition. be 4.1 Definition. Let (X, T) be a topological space, and let {Ai}zl a sequenceof subsetsof X. We define the limit inferior of {Ai}z=, , denoted

L-CONVERGENCE, by lim inf Ai, and the limit superior follows (illustrated in Figure 3):

TV-CONVERGENCE

21

of {Ai}z”=, , denoted by lim sup A,, as

(1) lim inf Ai = {x E X: for any U E T such that z E U, U all but finitely many i};

n Ai # 0 for

(2) limsupA,={x~X:foranyU~TsuchthatxEU,UnAi#Ofor infinitely many i}. 4.2 Definition. Let (X,T) be a topological space, let {Ai}z, be a is Lsequence of subsets of X, and let A C X. We say that {Ai}zl convergent in X to A, which we denote by writing Lim Ai = A, provided that lim inf -4, = A = lim sup A,.

For example, let {Ai}& be the sequencedepicted in Figure 3; we see are each that {Ai}& is not L-convergent whereas {Asi}E”=, and {A~-l}z~ L-convergent (with different limits). At this time, you may find it beneficial to work some (or all) of the exercisesin 4.10-4.17. These exercises are directly concerned with the concepts that we just introduced, and their solutions do not depend on any material in the rest of the section.

liminf,limsup Figure 3

(4.1)

22

I. THE TOPOLOGY

FOR HYPERSPACES

We discuss some terminology and phraseology that we use in the theorems. Convergence in CL(X) means, of course, convergence with respect to the Vietoris topology TV. We call this convergence TV-convergence. We want to always remind ourselves of the inherent difference in perspective between TV-convergence and L-convergence: TV-convergence takes place in CL(X) whereas L-convergence takes place in X. We emphasize the and “L-convergence in difference by saying “TV-convergence in CL(X)” X.”

Relationships

between L-convergence TV-convergence

and

Our theorems are concerned with implications between TV-convergence and L-convergence of sequences.It would be awkward, if not distracting, to continually refer directly to sequencesand their limits in the statements of the theorems. Thus, we simply say that one type of convergence implies the converges to other, by which we mean the following: If a sequence,{Ai}El, A with respect to the first type of convergence, then the sequence {Ai}zl converges to A with respect to the second type of convergence. Note the following simple but useful lemma. 4.3 Lemma. Let (X, T) be a topological space. Let {A,}z, be a sequence of subsets of X, and let .4 c X. Then, (1) and (2) below are equivalent:

(1) Lim -4, = A; (2) A

c

lim inf Ai and lim sup Ai c A.

The lemma follows immediately from the definitions in 4.1 and 4.2, it being evident from 4.1 that lim inf Ai c iim sup A,. H Proof.

4.4 Theorem. Let (X,T) be a regular topological space. Then, TVconvergence in CL(X) implies L-convergence in X. Proof. Assume that {A,)zO=, is a sequencein CL(X) such that {Ai}p”l TV-converges to A in CL(X). We will show that Lim ‘4i = A by using 4.3. First, we show that A C lim inf Ai. Let a E A, and let U E T such that a E U. Note that A E (X,U). Thus, since {Ai}zO=, TL+converges to A, there exists N such that

Ai E (X, U) for all i 2 N;

RELATIONSHIPS

BETWEEN

L-CONVERGENCE

AND..

.

23

in other words, Ai n U # 8 for all i 2 N. This proves that a E lim inf Ai. Therefore, we have proved that (1) A c lim inf Ai. Next, we show that lim sup A, c A. Let CCE X - A. Since (X,T) is regular, there exist U,W E T such that z E U, A c W, and U n W = 0. Then, since A E (IV) and {Ai}:“=, T v - converges to A, there exists A4 such that Ai E (IV) for all i 2 M. Thus, since U n W = 0, we see that A, n U = 0 for all i 2 M. This shows that x $! lim sup A,. Therefore, since we started with any point x # A, we have proved that (2) lim sup Ai c A By (l), (2), and 4.3, Lim Ai = A. H We turn our attention to the converse of 4.4, that is, to determining when L-convergence implies TV-convergence. Of course, we must only consider L-convergence for sequences whose terms are in CL(X). Under this trivial but necessary restriction, we show that L-convergence implies TVconvergence when X is countably compact (4.6); we also show that this implication between convergences is sufficient for a Ti-space X to be countably compact (4.9). Recall that a topological space, X, is said to be countably compact provided that every countable, open cover of X has a finite subcover [5, p. 11. Let (X,T) be a countably compact topological space. If is a sequenceof nonempty subsets of X, then

4.5 Lemma.

{Ai}fZl

lim sup Ai # 0. Proof. First, let us note the following fact (whose proof we leave as an exercise in 4.11): (#) lim sup Ai = n:zp,l[cl(u~nA,)]. Next, let U, = X - cl(Ug=,A,) for each n = 1,2,. . .. Now, supposethat lim sup Ai = 0. Then, by (#), Ur=iU,, = X. Hence, (X,T) being countably compact, finitely many of the sets U,, cover X. Of these finitely many sets that cover X, let U,,, be the one having the largest

24

I. THE TOPOLOGY FOR HYPERSPACES

index m. Then, since U, c U,,, for each n < m, obviously means that +&A,) = 0. Therefore, lemma. #

in particular,

A, = 0. This contradicts

U,,, = X. This

an assumption

in the

4.6 Theorem. Let (X,T) be a countably compact topological space. Then, L-convergence in X for sequences in CL(X) implies TV-convergence in CL(S). be a sequence in CL(X) such that {Ai}zl is LProof. Let {Ai}gl convergent in X to a subset, A, of X (i.e., Lim Ai = A). First, we show that A E CL(X). By 4.2, A = lim sup Ai; hence, by 4.5, A # 0. The exercise in 4.12 shows that A is closed in X. Therefore, A E CL(X). is TV-convergent to A. Let Ur , , . . , U,, E NOW, we show that {Ai}gl T(n < co) such that

AE (&,...,Un). We must show that A, E (U,, . . . , U,,) for all sufficiently large i. We do this by finding integers, M and N, such that (1) and (2) below hold: (1) A, n U, # 0 for all i 2 M and each j = 1, . . . , n; (2) Ai c lJjn=rUj for all i 2 N. We find M so that (1) holds as follows. Consider any one of the sets U, by fixing j 5 n. Since A E (Ul, . . , U,), there is a point p E A fl U,. Since p E A and A = Lim Ai, we have by 4.2 that p E lim inf A,. Thus, since p E Uj and U, is open in X, we have by 4.1 that Uj fl A, # 0 for all but finitely many i. In other words, there is an integer, Mj, such that

Now, having obtained such an integer Mj for each j 5 n, we let

M = max {M1,...,Mn}. It is then evident that (1) holds for this integer M. Next, we show that there exists N such that (2) holds. For this purpose, let Y = x - uy&. Suppose, by way of obtaining a contradiction, that there is no integer N such that (2) holds. Then there is a subsequence, {Ai(k)}rzO=l, of {Ai}zl such that

WHEN X Is COMPACT HAUSDORFF (a) Ai

25

n Y # 0 for each k = 1,2,. . ..

We will use lim supy in denoting limit superiors with respect to the space (Y,TIY). Note that (Y,TIY) is countably compact (since Y is closed in X). Therefore, by (a), we may apply 4.5 to the space (Y,TIY) and to the sequence{Aick) n Y}p=i to conclude that (b) lim su~~(Ai(k) n Y) # 0 Note the following fact (which is evident from the definition in 4.1): (c) lim su~,(A~(~l

n Y) c lim sup Ai.

Now, since A = Lim Ai, we have by 4.2 that A = lim sup Ai. Hence, we see from (b) and (c) above that A n Y # 0; however, since A E (VI, . . . , U,J, clearly A c u$L~U~ and, hence, A n Y = 0. Thus, we have obtained a contradiction. Therefore, there must be an integer N such that (2) holds. By (1) and (2), Ai E (VI,. . . , U,J for all i 2 max {M, N}. Therefore, TV-converges to A. n we have proved that {Ai}zl

When

X Is Compact

Hausdorff

The following theorem is a consequenceof the two preceding theorems. 4.7 Theorem. Let (X,2’) be a compact Hausdorff space. Then, Lconvergence in X for sequencesin CL(X) .is equivalent to TV-convergence in CL(X). Proof. Since compact Hausdorff spacesare regular and countably compact, the theorem follows from 4.4 and 4.6. n 4.8 Corollary. Let (X,T) be a compact, metrizable space. Then, Lconvergence in X for sequencesin CL(X) is equivalent to convergence in CL(X) with respect to the Hausdorff metric. Proof.

The corollary follows from 4.7 and 3.2. I

An analogue of 4.8 for 2x when (X,T) is only metrizable is in Exercise 4.19. One incidental consequenceof 4.8 is worth noting: When (X,T) is a compact, metrizable space, then the Lim operator is the sequential closure operator for the Vietoris topology for CL(X). In contrast, when (X,T) is only metrizable, Lim may not be a sequential closure operator for any topology for CL(X) or, even, Fs(X) - seeExercise 4.20.

26

I. THE TOPOLOGY

Countable

FOR HYPERSPACES

Compactness

Is Necessary

We prove that the condition of countable compactness in 4.6 can not be weakened when (X, T) is a Tr-space. In fact, we prove even more: We restrict ourselves to F2(X) and we restrict ourselves to sequencesthat have a nonempty Lim. In the interest of conciseness,let us agree for the moment to call a sequence, {Ai}cl, of subsetsof X a nontrivial sequenceprovided that Lim A, # 0. 4.9 Theorem. Let (X,T) be a Ti-space. If L-convergence in X for nontrivial sequencesin FZ(X) implies TV-convergence in Fz(X), then (X, T) is countably compact. Proof. Suppose that (X,T) is not countably compact. Then, since (X, T) is a Ti-space, there is a countably infinite subset, Y, of X such that Y has no limit point in X [l, p. 2291. We assumethat Y # X (as we may, by removing a point from Y if necessary), and we let p E X - Y. Now, let yi for i = 1,2, . . . be a one-to-one indexing of the points of Y. For each i=l,2,...,let A

= {~,a).

We seeeasily that Lim A, = {p} (if z # p then, since z is not a limit point of Y and (X,T) is a Ti-space, there exists U E T with x E U such that is a nontrivial sequence U n A, # 0 for at most one i). Hence, {Ai}zl in Fs(X) with Lim Ai = (p}. However, the sequence { Ai}gi does not TV-converge in Fz(X) to {p} since {p} E (X - Y) but Ai 4 (X - Y) for any i (note that (X - Y) E TV because Y is closed in X). H Some research questions related to material in this section and the preceding section are in 83.1-83.5.

Exercises 4.10 Exercise. Let (X,T) be a topological space, and let What is Lim Ai when A, = A for each i = 1,2,. . .? 4.11 Exercise. i = 1,2,. . ., then

If (X,T)

A

C

X.

is a topological space and Ai C X for each

lim sup Ai = nr=i(cl(U&Ai)]. Remark. Regarding the formula for lim sup Ai in 4.11, there can not be a similar type of formula for lim inf Ai [2]!

EXERCISES

27

4.12 Exercise. Let (X,T) be a topological space, and let Ai C X for each i = 1,2,. . .. Then, lim sup Ai and lim inf A, are each closed in X; hence, assuming that Lim Ai exists, Lim Ai is closed in X. 4.13 Exercise. Let (X,T) be a topological each i = 1,2,. . .. Then . . lim mf Ai = hm mf A,, lim sup Ai = lim sup x2, and, assuming that Lim Ai exists, Lim A, = Lim xi. 4.14 Exercise. i = 1,2,. . ., then

space, and let A, c X for

If (X, T) is a topological spaceand A,, Bi C X for each

lim sup (Ai U &) = (lim sup Ai) U (lim sup Bi). Is the analogous formula for limit inferiors also valid? (Compare with the result in the next exercise.) 4.15 Exercise. foreachi=1,2,.... exists and, in fact,

Let (X,T) be a topological space, and let Ai, B, c X If Lim Ai and Lim Bi each exist, then Lim (A, U B2)

Lim (A, u &) = (Lim Ai) U (Lim Bi). 4.16 each i = If A1 If AI

Exercise. Let (X,T) be a topological space, and let Ai C X for 1,2, . . .. c Ax c ..., then Lim Ai = cZ(UEiAi). > AZ > ..., then Lim Ai = ~I~~cZ(A~).

4.17 Exercise. If {Ai}g”=, is a sequenceof connected subsetsof R’ and Lim A, exists, then Lim Ai is connected. Give an example of a sequence, {Ai}fZO=,, of compact, connected subsets of a metric space, X, such that Lim Ai exists, Lim Ai is compact, but Lim Ai is not connected. 4.18 Exercise. Let (X,T) be a compact Hausdorff space. Then, Lconvergence in X for sequencesin C(X) is equivalent to TV-convergence in C(X). Hence, when (X, T) is compact and metrizable, L-convergence in X for sequencesin C(X) is equivalent to convergence in C(X) with respect to the Hausdorff metric.

28

I. THE

TOPOLOGY

FOR HYPERSPACES

4.19 Exercise. Let (X, T) be a metrizable topological space. If { Ai}zI is a sequence in 2x such that Lim Ai E 2x, then {Ai}gI converges with respect to the Hausdorff metric to Lim Ai. Conversely, convergence in 2X with respect to the Hausdorff metric implies L-convergence in X. 4.20 Exercise. The purpose of this exercise is to show that the Lim operator is not necessarily a sequential closure operator for any topology for Fs(X) when the space (X, T) is metrizable (recall the discussion in the second paragraph after the proof of 4.8). First, let us define what it means for Lim to be a sequential closure operator. Let (X,T) be a topological space. Let ?/ C CL(X). For any A C ?t, define L(d) as follows: L(d) = {B E 3-1: B = Lim Ai for some sequence{Ai}c”=, in A}. Then, saying “Lim is a sequential closure operator for some topology for 3-1” simply means that the function L is a closure operator. (It is easy to see that L always satisfies the first three axioms for closure operators in [4, p. 381; hence, L is a closure operator when L is idempotent, i.e., when L 0 L = L.) Now, for our example. Let X be the subspace of R’ consisting of the following numbers: 0; 1; p, = 1+ (l/n) for n = 1,2,. . .; & = (l/n) + (l/k) for n = 2,3,. . ., k = 1,2,. . ., and for which qk < l/(n - 1). Prove that Lim do this by is not a sequential closure operator for any topology for F2(X); considering the subset A of F2(X) given by d={{p,,qL}:n=2,3

,...

andk=l,2

,... }.

References 1.

James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1967 (third printing).

2.

R. Engelking, Sur l’impossibilite’ de dt?j%ir la limite topologique infkrieure ci l’aide des ope’rations dt!nombrables de l’algtbre de Boole et de l’ope’ration de fermeture, Bull. Acad. Polon. Sci. Cl. III 4 (1956), 659-662; M.R. 19 (1958), 668.

3. 4.

F. Hausdorff, Mengenlehre, Walter de Gruyter & Co., Berlin, 1927. K. Kuratowski, Topology, Vol. I, Acad. Press, New York, N.Y., 1966.

5. 6.

K. Kuratowski, Topology, Vol. II, Acad. Press, New York, N.Y., 1968. Ernest Michael, Topologies on spacesof subsets,Trans. Amer. Math. Sot. 71 (1951), 152-182.

REFERENCES 7.

8. 9.

29

Sam B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. R.E. Smithson, First countable hyperspaces, Proc. Amer. Math. Sot. 56 (1976), 325-328. Daniel E. Wulbert, Subsets of first countable spaces, Proc. Amer. Math. Sot. 19 (1968), 1273-1277.

This Page Intentionally Left Blank

II. Examples: Geometric Models Hyperspaces

for

A geometric model for a hyperspace is a picture that shows what the hyperspace looks like. We obtain many geometric models for hyperspaces. When natural limitations prevent us from drawing a picture of the entire hyperspace, the geometric model will contain enough information to give us a clear mental image of the hyperspace. All of our spaces X are nonempty, compact, and metric. We call a nonempty, compact, metrizable space a compactum; we call a connected compactum a continuum [24]. We use the words subcompactum and subcontinuum when referring to a compactum and a continuum (respectively) as being a subset of a space. According to the terminology just introduced, 2x is the hyperspace of all subcompacta of the compactum X, and C(X) is the hyperspace of all subcontinua of X. All of our geometric models are for these two hyperspaces. In fact, most of our geometric models are for C(X) (the exception being in section 8). Thus, 2” plays a minor role in this chapter; however, 2x plays a major role in Chapter III, which is in reality a continuation of the study of geometric models. Let us note some pertinent information from Chapter I. Let X be a compactum. Then, 2” and C(X) are also compacta (by 3.1, 3.5, and 3.7). We will be concerned with proving that various functions defined on (subspaces of) 2” or C(X) are continuous. Since 2x and C(X) are metrizable (by 3.1), we can use sequences in proving that the functions are continuous. With this in mind, we recall that for sequences in 2x or C(X), TV-convergence, Hausdorff metric convergence, and L-convergence in X are equivalent to each other (by 4.7, 4.8, and Exercise 4.18). 31

32

II. EXAMPLES:

GEOMETRIC

MODELS

FOR HYPERSPACES

We take this opportunity to give somegeneral terminology and notation that we use throughout the chapter. We use the term nondegenerate when referring to a space t,o mean that the space has at least two points. We use II or x in denoting Cartesian products. We tacitly assumethat all Cartesian products have the Tychonoff (product) topology [18, p. 1471. We will frequently be concerned with Cartesian products of intervals, most often when all the intervals are [O,l]. We use [0, 11% to denote the ith coordinate factor for such Cartesian products. Now, for n = 1,2,. . ., an n-cell is a space that is homeomorphic to I” = IIy==,[O, l]i. A l-cell is called an arc; an end point of an arc, A, is either one of the two points of A that are the image of the end points of [O,l] under any homeomorphism of [O,l] onto A. For each n = 1,2,. . ., aP denotes the manifold boundary of In; in other words, dI” = ((~&

E I” : 5, = 0 or 1 for some i).

A space that is homeomorphic to aP is called an (n- l)-sphere. A l-sphere is called a simple closed curve. Any n-cell is an n-manifold with boundary; we denote the manifold boundary of any n-cell, 2, by dZ. A Hilbert cube is a space that is homeomorphic to I* = llfZo=,[0, l]i. We note that the standard metric, d,, for I” is defined as follows [18, p. 2121: dm((s,)~l,

(yi&)

= 2

2-‘1xi -

yil forall (zi)El, (yi)El E P.

i=l

The dimension of a spacemeans the topological dimension of the space [9]. We mostly use dimension in describing geometric objects - usually polyhedra - for which dimension has a clear geometric meaning. The only time that we use dimension extensively in a technical way is in section 8, where we are concerned with zero-dimensional spaces (we define this notion near the beginning of section 8). We often use the term “finitedimensional”, which the reader can interpret as meaning that the space being referred to is embeddable in R” for some n. The term “infinitedimensional” means not finite-dimensional. We denote the dimension of a space, Y, by dim(Y). Finally, it will be helpful in building many of our geometric models to consider the following subspacesof 2x and C(X). Let X be a compactum, and let K be a subcompacturn of X. We define 25 and CK(X) as follows: 2; = {A E 2x : A > K},

CK(X)

= {A E C(X) : A 3 K}.

We refer to 2: and CK(X) as containment hyperspaces;more precisely, 2; is the containment hyperspacefor K in 2x, and CK(X) is the containment

5. C(X)

FOR CERTAIN FINITE GRAPHS

X

33

&per-space for K in C(X). For a point p E X, we write 2: and C,(X) for the containment hyperspaces for (p}. We note that since 2x and C(X) are compacta, containment hyperspaces are compacta (by Exercise 1.19).

5.

C(X)

for Certain

Finite

Graphs

X

A finite graph is a continuum that can be written as the union of finitely many arcs, any two of which can intersect in at most one or both of their end points. Alternatively, a finite graph is a compact, connected polyhedron of dimension zero or one (the zero-dimensional casebeing the polyhedron that consists of only one point). We construct geometric models for C(X) when X is an arc, a simple closed curve, a noose, and a simple n-od. Regarding 2x when X is a finite graph, seeChapter III.

X an Arc 5.1 Example. We construct a geometric model for C(X) when X is any arc. Specifically, we show that C(X) is a 2-cell. In addition, we determine which subcontinua of X form the manifold boundary of C(X). We first consider the case when X = [0, 11. Observe that the points of C([O,l]) are the closed intervals [a, b] withOO) = f(K)

under this assumption, we have from the formula for f that (v(A),@

=

(v’(BK),tK

-

1);

hence, tK = 1, which means that BK = K; thus, (cp(A),O) = (v(K), Oh therefore, finally, A = K (since cpis one-to-one). Now, since g is one-to-one and since ‘p and f are homeomorphisms on the compact spacesC(S’) and Ci (X) (respectively), we seethat g is a homeomorphism of C(S’) U Cl(X) into R3. We need two specific facts about g - (2) and (3) below - for use in = Step 3. Recall from Step 1 that (p[C(S’)] = D; thus, since f[Cl(X)] D’ x [0, 11, we have that (2) g[C(S’) u C,(X)] = (D x (0)) u CD’ x [O, 11). Next, by the formula for f and by (1) near the end of Step 1, observe that f([l,t])

= ((l,O),t

- 1) for all [l,t]

E Cl(J);

thus, since g(Ci(J) = f]Ci(J), we have that (3) g([l,t]) = ((l,O),t - 1) for all [l,t] E Cl(J). Step 3: Extending g to a homeomorphism, h, of C(X) onto the polyhedron in Figure 6, p. 37. Denote the polyhedron in Figure 6 by P. Let

X A SIMPLE

n-oD

39

T denote the triangular 2-cell in P with vertices (( 1, 0), 0), ((l,O), l), and ((2,0), 1). Let K = cl(P - T). By (2) of Step 2, g maps C(S’) U Ci (X) homeomorphically onto K. We also note the following three relevant facts: C(X) = C(S’) u Cl(X) u C(J), [CW)

u

Cl(X)]

n

C(J)

=

Cl(J),

and, by (3) at the end of Step 2, T n K = g[C, (cl)]. Thus, to obtain the homeomorphism h of C(X) onto P that we want, it follows that we only need to find a homeomorphism, j, of C(J) onto T such that j]Ci(J) = g]Ci(J). Th is is easy to do. Define j : C(J) + T by letting j([s, t]) = ((s, 0), t - 1) for all [s, t] E C(J). We seethat j satisfiesour requirements: j is a homeomorphism of C(J) onto T (J’ being a simple modification of the homeomorphism that we defined in 5.1), and j]Ci(J) = g(Ci(J) by (3) of Step 2. Therefore, letting h(A) =

g(A),

if A E C(S’) U Cl(X)

j(A),

if A E C(J)

we seethat h is a homeomorphism of C(X) onto the polyhedron in Figure 6, p. 37. n

X a Simple

n-od

In 5.4 we construct a geometric model for C(X) when X is any simple n-od. Note the following terminology. A simple n-od (n > 3) is a finite graph that is the union of n arcs emanating from a single point, V, and otherwise disjoint from one another (Figure 7, top of the next page). The point 2, is called the vertex of the simple n-od, and each of the n arcs is called a spoke of the simple n-od. A simple 3-od is called a simple triod. A standard simple n-od is a simple n-od lying in some metric linear space such that each spoke of the simple n-od is a straight line segment. An n-fin (n > 3) is a continuum that is the union of n 2-cells (called fins) all of which intersect in a single point and any two of which intersect only in that point (Figure 8, bottom of the next page). Any n-fin, F, evidently contains a simple n-od any two of whose spokeslie in the manifold boundaries of different fins of F; we call a simple n-od that is situated this way in F a base of the n-fin F. (The use of the word “base” comes from thinking of an n-fin in the following way: If Y is a simple n-od with vertex U,

40

II. EXAMPLES:

GEOMETRIC

MODELS

Simple n-od Figure 7

n-fin Figure 8

FOR HYPERSPACES

X A SIMPLE

n-OD

41

then the decomposition spaceof Y x [0, l] obtained by shrinking {v} x [0, 11 to a point is an n-fin; and Y x (0) is a base of this n-fin.) 5.4 Example. We show that when X is a simple n-o& a geometric model for C(X) is the n-dimensional polyhedron that is represented in Figure 9 (top of the next page). The polyhedron may be described as follows: It is the result of attaching an n-fin to the n-cell I” by identifying a base of the n-fin with a standard simple n-od lying in dI”. To be more precise, let F,, denote an n-fin, and let Y be a standard simple n-od in dI”; then, the polyhedron that Figure 9 is intended to suggest is a quotient space - it is the attaching space F, Uf I” that is obtained from the free (disjoint) union of F, and I” by means of a homeomorphism f of a base of F,, onto Y. Regarding our requirement that Y be a standard simple n-od, see 5.5. (For basic information about attaching spaces,see, e.g., [7, p. 1271 or [34, p. 651.) For the proof that C(X) M F, Uf I” when X is a simple n-od, we denote the vertex of X by v and the spokes of X by S1, . . . , S,. The proof focuses on two dominant parts of C(X), namely, the containment hyperspace &(X) and 3 = U~==,C(S~,). Note that C(X) = C,(X) u 3. Also, note that C,(X) n 3 = Ur==, C,(Si), which we denote by f?. We prove first that CV(X) is an n-cell; in fact, as is more pertinent, we define a homeomorphism, (p, of C,(X) onto I” such that (p(B) is a standard simple n-od in dI”. We define ‘p in terms of its coordinate functions . , (Pi. For each i < n, let & be a homeomorphism of Si onto [O,l] such ~~~‘[i(v) = 0; then de&e (Pi : Cv(X) + [0, l] as follows: pi(A) = sup[&(A n Si)] for each A E C,,(X). It follows readily that each (pi is continuous (verify first that the function that assignsA n Si to each A E C,(X) is continuous). Clearly, each pi maps CV(X) onto [O,l]. Also, the family of functions (PI,. . . , I++ separates points: if A, B E C,,(X) such that A # B, then clearly A n Sj # B n Sj for somej; hence, it follows easily that cpj(A) # ‘pj (B). Therefore, we now seethat cp= (cpl,... , cp,) is a homeomorphism of C,,(X) onto I”. Finally, cp(L?)is a standard simple n-od in 81” since v(B) = Y, where Y = {(ti)F=l E I” : ti = 0 for all but at most one i}. Next, we prove that 3 is an n-fin and that B is a base of 3. To seethat 3 is an n-fin, simply note that each C(Si) is a 2-cell (by 5.1 .l) and that C(Si) n C(S,) = {{v}>

whenever i # j.

42

II. EXAMPLES:

GEOMETRIC

C(X),X

MODELS

FOR HYPERSPACES

a simple n-od (5.4) Figure 9

To seethat f? is a base of the n-fin .T, we make three observations: (1) C,,(S) is an arc for each i (cf. 5.1.1); (2) C,(Si) n C,(Sj) = {{u}}

whenever i # j;

(3) C,(Si) C LW(Si) for each i (cf. 5.1.1). BY (1) and (21, B is a simple n-od with vertex {w} and spokes C,,(S); by (3), the spokes of B lie in the manifold boundaries of different fins of T. Therefore, f3 is a base of FT. On the basis of what we have shown, it is easy to see why C(X) M F, Uf In. Nevertheless, we include a proof. For convenience, we choose F, to be F’; we assumethat 3 rl In = 0 (so that we may form the attaching space F,, Uf I” with F,, = 3). We then let f = cp[L?,which maps B onto Y C I”. Before proceeding directly with the proof, we make a few preparatory comments about the nature of the points of F, Uf I”. The points of F,, Uf I” are equivalence classesof points of the disjoint

X

A SIMPLE

43

n-OD

union F, U I”. The relevant equivalence relation, -, on F, U In is the one generated by declaring that B N f(B) for each B E B (recall that a c F,, since F, = F). Therefore, since f = (~]a is one-to-one, the only nondegenerate equivalence classesare {B, p(B)} for the points B E L?. We adopt the following notation for the points of F, Uf I”: for any z E F, UP, [z]- denotes the equivalence classwith respect to - that contains z. Thus, (#)

[B], = {B,p(B)}

= [p(B)],

whenever B E L3,

and [z]~ = { z } whenever z E F, U In - (B U q(D)). In light of the comments just made (especially (#)), we see that the following formula gives a well-defined function, h, from C(X) to F, Uf 1” (recall that f = ‘p]L?): h(A) =

M-1 b(A)l-

ifAEF 7 if A E C,,(X)

Therefore, since ‘p is a homeomorphism of C,,(X) onto In, it follows easily that h is a homeomorphism of C(X) onto F,, Uf I”. Furthermore, LJ is a base of the n-fin 3 = F,,, the attaching map f is a homeomorphism of I3 onto Y, and Y is a standard simple n-od in d1”. n 5.5 Remark. We used the standard simple n-od Y in df” to construct a geometric model for C(X) in 5.4. This resulted in an especially clear geometric model - and, after all, that is our goal. Nevertheless, we make three comments about our choice of Y in 5.4 (reasons that justify the comments will follow) : (1) we could not have let Y be any simple n-od in dI” when n > 4; (2) we could have only required that Y be tame in 81” (definition follows); (3) when X is a simple triod and Y is any simple triod in 813, then C(X) M F3 Uf I3 (where F3 and f are as in 5.4). Before we justify the comments, we give a definition for the terminology in (2). A simple n-od, 2, in i?In is said to be tame in df” provided that there is a homeomorphism, h, of 81” onto OI” such that h(Z) is a standard simple n-od; otherwise, Z is said to be wild in dI” [33]. Now, concerning the comment in (l), there are wild simple n-ods in 81” for any n 2 4 (since there are wild arcs in dP for n 2 4 [33, p. 841). On the other hand, Y in 5.4 is tame in 81”; thus, since the homeomorphism cp in 5.4 from Cv(X) onto I” takes Z?onto Y, we seethat D is tame in &7,(X). Therefore, C(X) could not possibly be homeomorphic to F, Uf I” if f were a homeomorphism of D onto a wild simple n-od in 81”. This verifies (1). The comment in (2) is evident from the definition of tame and from the

44

II. EXAMPLES:

GEOMETRIC

MODELS

FOR HYPERSPACES

fact that the simple n-od B in 5.4 is tame in G’Cv(X). Finally, (3) follows from (2) since any simple triod in 131~is tame in d13 (which can be proved using the Schoenflies Theorem [33, p. 471). More about finite graphs is in section 65 and in the exercises at the end of section 72.

Historical

Comments

Duda did an in-depth study of C(X) when X is a finite graph ([4]-[6]). He provided a lot of specific information that can be used in constructing geometric models for C(X), and he constructed several such models (see, especially, [6, pp. 248-2551). He also obtained general results, two of which are particularly relevant to what we have done: A continuum, X, is a finite graph if and only if C(X) is a polyhedron [4, p. 2761; if P is a polyhedron and dim(P) 2 3, then P NN C(X) for at most one finite graph X [4, p. 2831. Earlier, Kelley had proved that if X is a Peano continuum, then dim [C(X)] < 00 if and only if X is a finite graph [16, p. 301. Furthermore, Kelley provided a formula for calculating dim [C(X)] when X is a finite graph [16, P. 301 (seethe Remarks in [4, p. 2781). We mention that Kelley stated (without proof) the “only if” part of the first result of Duda quoted above (see [16, p. 311). Recent developments concerning C(X) when X is a finite graph are mostly concerned with Whitney levels (e.g., see [ll] and [13]-[15]; we discusssome of these results in section 65).

Exercises 5.6 Exercise. Construct a geometric model for C(X) when X is a figure eight. (A figure eight is a finite graph consisting of two simple closed curves that intersect in a single point.) 5.7 Exercise. Construct a geometric model for C(X) when X = S’ U 2, where S1 is the unit circle in R2 and Z is the straight line segment from Kw) to GAO). 5.8 Exercise. Construct a geometric model for C(X) when X = S’ U Y U Z, where S1 is the unit circle in R2, Y is the straight line segment from (1,0) to (2,0), and 2 is the straight line segment from (-1, 0) to (-2, 0). 5.9 Exercise. What is dim [C(X)] when X is the finite graph drawn in Figure 10 (top of the next page)? In other words, what is the largest n such that C(X) contains an n-cell? 5.10 Exercise.

Let X be a finite graph.

45

EXERCISES

x for 5.9 Figure IO

If C(X) is a 2-cell, then X is an arc or a simple closed curve (and conversely). Can C(X) be an n-cell for n > 2? If C(X) is the polyhedron in Figure 6, p. 37, then X is a noose. (Prove this without using Duda’s theorem [4, p. 2831, which we stated in the last paragraph of the section.) 5.11 Exercise. Let X be an arc with end points p and q. In the proof of 5.1.1, we used k’ to show that

K(X)

= Fl (X) u C,(X)

u C,(X).

Avoid the use of k* by proving the following fact directly: If A E C(X), then A has an open neighborhood that is homeomorphic to R2 if and only if A 4 Fi(X), p $! A, and q 4 A. 5.12 Exercise. The purpose of this exercise is to gain some insight into a new concept; we will use the concept in connection with geometric models in the next exercise.

46

II. EXAMPLES: GEOMETRIC MODELS

FOR HYPERSPACES

Let A be a collection of nonempty sets. A choice function for A is a function f : A 4 ud such that f(A) E A for each A E A. Choice functions are often called selections when they are continuous on a hyperspace. The Axiom of Choice says that choice functions for A always exist [12]. Thus, there is a choice function for any hyperspace. However, the Axiom of Choice says nothing about the continuity of a choice function for a hyperspace: (1) Let I = [0, 11. Find two selections for 2’. Prove that these are the only two selections for 2’. (2) Find a selection for C(X) when X is any n-od. Regarding (1) and the change to C(X) in (2), we remark that the arc is the only continuum X for which there is a selection for 2x. In fact, the arc is the only continuum X for which there is a selection for Fz(X). These results are in [19] and can be deduced from 1.9 and 7.6 of [22]; see Exercise 75.19. 5.13 Exercise. We give the application of geometric models of hyperspacesthat we mentioned after the proof of 5.2. Prove that there is no continuous choice function for C(S’) by using the geometric model for C(S’) in 5.2. [Hint: Use a form of the Brouwer fixed point theorem.] Remark. For applications of the result in 5.13 and for more about selections, seesection 75 and [25, pp. 253-2671. 5.14 Exercise. Let h : C(S’) + D be the homeomorphism that is defined in the proof of 5.2. Let p = (l,O). What is the precise shape of the image of the containment hyperspace C,(S1) under h? What medicalmathematical phraseology might appropriately describe removing the interior of h[C,(Sl)] from D?

6.

C(X)

When

X Is the Hairy

Point

Our previous geometric models are for finite-dimensional hyperspaces C(X). We now construct a geometric model for C(X) when C(X) is infinitedimensional. Specifically, we construct a geometric model for C(X) when X is the hairy point. We depict the hairy point in Figure 11 (top of the next page). It is a natural, infinite extension of simple n-ods that can be defined as follows. The hairy point is a continuum that is the union of countably infinitely many arcs HI, Hz,. . . satisfying the following conditions: All the arcs Hi

6. C(X)

WHEN X Is THE HAIRY POINT

Hairy point Figure 11

co-fin Figure 12

47

48

II. EXAMPLES:

GEOMETRIC

MODELS

FOR HYPERSPACES

emanate from a single point, v, and are otherwise disjoint from one another, and limi+, diameter (Hi) = 0. Each of the arcs Hi is called a hair, and the point v is called the follicle of the hairy point. We use the following terminology, which is an extension of terminology preceding 5.4. An co-fin (Figure 12, bottom of the previous page) is a continuum that is the union of countably infinitely many P-cells Fl , F2, . . . all of which intersect in a single point and any two of which intersect only in that point, and such that limi+, diameter (Fi) = 0. Each of the 2-cells Fi is called a fin. Any m-fin, F, obviously contains a hairy point any two of whosehairs lie in the manifold boundaries of different fins of F; we call a hairy point that is situated this way in F a base of the oo-fin F. We now give the example for this section. 6.1 Example. Let X be the hairy point. We obtain a geometric model for C(X) as follows (the model is drawn in Figure 13, top of the next page): We assumethat the Hilbert cube I” has its standard metric d, (defined near the beginning of the chapter). Let Y = {(t%)zr E I”

: ti = 0 for all but at most one i}.

Note that Y is a hairy point, its J‘th hair being the points of Y such that t, = 0 for all i # j (the da-diameter of the jth hair is 2-j). Next, let F, be an oa-fin such that F, n IO0 = 0. Finally, let f be a homeomorphism of a base of Foe onto Y. We show that C(X) x F, uj I=‘=, where F, Uf Iw is the attaching space obtained from F, U I” by means of the attaching map f ([7, p. 1271or [34, p. 651). The proof that C(X) M F, Uf P is similar to the proof in 5.4. We sketch the proof, omitting the details of the verifications (since the details are straightforward modifications of the details in 5.4). Let v denote the follicle of X, and let H1, Hz,. . . denote the hairs of X. Let T = UgiC(Hi), and let B = C,(X) n FT. Note that C(X) = C,(X)

u FT.

6. C(X)

WHEN

X Is THE HAIRY

POINT

49

C(X), X the hairy point (6.1) Figure 13

We show first that C,(X) is a Hilbert cube; in fact, we define a homeomorphism, (p, of C,(X) onto I” such that cp(B) = Y (where Y is as defined above). We define the coordinate functions, vi, of ‘p as follows: for each i= 1,2,..., let & be a homeomorphism of Hi onto [O,l] such that h aving p as an end point. Let X be a continuum, and let A E C(X). Then, A is order arcwise accessiblefrom 2x - C(X) if and only if A # Fr(X). Remark. The result in 15.18 shows that each A E C(X) - Fl(X) is arcwise accessiblefrom 2.’ - C(X); nevertheless, a singleton may not be arcwise accessiblefrom 2x - C(X) (for examples, see [27, pp. 10 and 131 or [29, pp. 378 and 3831). 15.19 Exercise. Show that our theorems about homogeneous hyperspaces in 15.6 and 15.7 do not generalize to Hausdorff continua by giving examples of the following: A locally connected Hausdorff continuum, S, such that 2x is not homogeneous;a locally connected Hausdorff continuum, X, with no free generalized arc such that C(S) is not homogeneous.

16.

Kelley’s

Segments

We define Kelley’s notion of a segment [16], and we establish the basic properties of segments. This provides a foundation for the next section, where the results and ideas that we present here take on new life in a dynamic setting.

Kelley’s

Notion

of a Segment

There is a natural and especially useful way to parameterize any order arc. The method involves Whitney maps. Let us see how it is done. This will lead to Kelley’s notion of a segment in 16.1. Let X be a compactum, and let w be a Whitney map for 2”. Let cy be an order arc in 2x from A0 to Al. Let ae = w(AO) and let al = w(Al). Then, W((Yis a homeomorphism of 0: onto [aa,ar] (by 14.2). Let X denote the linear map of [0, 11onto [a~, al] given by x(t) = (1 - t) . a0 + t. al for all t E [O,l]. Finally, define the parameterization u from [O, l] onto cy by letting 0 = (wla)-’

0 A.

Now, having defined the parameterization U, let us determine some of the elementary properties of o. Our first two properties of 0 are rather evident: o is a homeomorphism of [0, l] onto cu; a(0) = A0 and o(l) = A1 (since a(O) = (~]a)-‘[X(O)] = (wlcr-‘(ao) = A 0, and similarly for a(l)).

128

IV. ARCS IN HYPERSPACES

Our next property of CJis that ‘w linearizes c - explicitly, w o 0 = A (which is obvious). We prefer to express the fact that w o c = X without mentioning XT as follows: for each t E [0, 11, w[a(t)]

= (1 - t> . w[a(O)]

+ t . w[a(l)].

The right-hand side of the equality is, indeed, A(t) since w[a(O)] as and w[o(l)] = w(A1) = al. Our final property of g is that ~7is order preserving: rr(ti)

c

o(tz) whenever

05

tl

= w(A,,)

=

0 such that if dy(ylryz) < 6, then &(fh),

f (yz)) < E for all f E .K

COMPACTNESS

135

It is easy to see that since Y and Z are compacta, the notion of equicontinuity depends only on the topologies on I’ and 2 (and not on the given metrics dy and dz). 17.2 Arzela-Ascoli Theorem (special case). Let E’ and 2 be compacta, and let F C 2’. Then, 3 is compact if and only if F is equicontinuous and F is closed in 2’.

Proofs of the Arzela-Ascoli Theorem are in many texts; for example, see [6, p. 2671or [15, p. 2331. We use the following lemma in the proof of our theorem about, compactness of spacesof segments. At first, the lemma may appear to have no direct bearing on the compactness of spaces of segments ~ however, the lemma essentially proves that spacesof segmentsare equicontinuous. 17.3 Lemma. Let X be a compactum, let ‘R be a closed subset,of 25, and let w be a Whitney map for 31. Then, for any given E > 0, there is an Q(E) > 0 with the following property: If A, B E 3c such that, A c B and Iw(B) - w(A)1 < T(E), then H(A,B) < E. Proof. Suppose that, the lemma is false for some particular E > 0. Then, for each i = 1,2,. . ., there exist Ai, Bi E fl such that. iii c B;. lw(Bi) - w(A,)I < t, and H(Ai, B,) 2 E. Since 7-t is a compacturn (by 3.1 and 3.5), we can assumethat the sequences{A,}z”=, and {Bi};M=, converge in 31 to, say, A and B, respectively. Then we seethat A C B and w(A) = w(B); hence, by (1) of 13.1, A = B. However, since H(Ai, Bi) 2 E for each i, we also have that H(A, B) > E. Therefore, we have a contradiction. n 17.4

Let X be a compactum, and let 3c be a closed subset is compact.

Theorem.

of 2X. Then, S,(Z)

Proof. We show that S,(R) is equicontinuous and closed in ~[‘,‘l; then our theorem follows from 17.2 (since Fl, being closed in 2x, is a compactum). To prove that S,(X) is equicontinuous, let E > 0. Let q(~) be as guaranteed by 17.3. Let s = sup w(R). Choose 6 > 0 such t,hat 6.9 < V(E). Now, let tl, ta E [0, l] such that It1 - tzl < 6, and let D E S,,(R). We see at once from (3) of 16.1 that w[a(h)]

-

w[c(h)]

=

(t2

-

tl)

. w[a(O)]

-

(t2

-

t1)

. w[a(l)].

Hence, I+(tl)l

-

wb(tz)]l

=

It;?

5

1t.L -

-

t1 I t1I

Iw[40)] . s
p(ar,as). Then, by the way p is defined, H(al(t), an(t)) < r for each t E [O,l].

Thus,n([O, 11)~NH(f,~2([0,11))

and

02([0,11)

c

Nd~,m([o,

11)).

Hence,

by the first part of Exercise 2.9,

Thus, we have proved that HH(~~((T~), fw(cn)) < T for any r > p(ar,az). Therefore, we have proved (*). Clearly, (*) implies that fw is continuous. Finally, we prove that f;’ : B(X) + S,(R) is continuous. Let a E D(X), and let cy( E B(X) for each i = 1,2,. . . such that the sequence {oi}gr converges in ??(‘?I) to a. Let (T= f;‘(cu), and let oi = f;‘(ai) for each i. We prove the following fact: (#) There is a compact subset, C, of &,(‘?I) such that u E C and gi E C for each i.

138

IV. ARCS IN HYPERSPACES

To prove (#), let %’ = (Y U (U&ai), Whitney map for 3t’), and let

let w’ = ~1%’ (note that w’ is a

c = s,!(w); we show that C satisfies the conditions in (#). An easy sequenceargument shows that U’ is compact; hence, by 17.4, C is compact. Since 20’ = wllfl’ and 31’ C ?t, we seeeasily that c = {a’ E S,(U)

: o’([O, 11) c U’};

thus, C c SW(Y), cr E C, and pi E C for each i. This completes the proof of (#I. NOW, we use (#) to show that {f;‘(~i)}~r converges in S,,,(‘?Q to f;‘(a). Recall th at we have proved that fw is one-to-one and continuous on S,,,(U);hence, by (#), we have that (1) fw]C is a homeomorphism. Next, recall that o = f;‘(a) and pi = f;l(czi) for each i; hence, by (#), we have that (2) cr E fw(E) and cyi E fw(C) for each i. Finally, recall that {ai)E1 converges to a; hence, by (1) and (a), {f,~‘(cy~)}~~ converges in C to f;‘(a). Thus, since C C S,(x) by (#), we have shown that {f;l(cyi)}p”,I converges in S, (3t) to f;’ (a). Therefore, we have proved that f;’ is continuous. m Usually, SW,(Ufl) # SW,CW w hen w1 # w2 (e.g., see Exercise 16.12). Nevertheless, we have the following result: 17.8 Corollary. Let X be a compactum, and let ‘fl c 2x. Then, for (‘IY) zz SW,(31). any two Whitney maps wi and w2 for ?i, SW, Proof.

By 17.7, S,, (31) = 8(R)

S,(Zx),

S,(C(X))

When

and S,,(x)

M B(x).

X Is a Peano

n

Continuum

We prove a theorem about spacesof segments that is analogous to the Curtis-Schori Theorem in 11.3. Let X be a nondegenerate Peano continuum. Then (1) S,(2x) is the Hilbert cube, (2) &,(C(X)) is the Hilbert cube when there is no free arc in X, and (3) S,(C(X)) is an AR (i.e., a Hilbert cube factor by 11.4). 17.9

Theorem.

S,(Zx),S,(C(X))

WHEN X Is A PEANO CONTINUUM

139

Proof. By 17.7, we may as well prove the theorem for ??(2x) and n(C(X)); we base the proof on Toruriczyk’s Theorem in 9.3; we use ideas in section 11 to verify the assumptions in 9.3. The proof that ?!?(2x) and B(C(X)) are absolute retracts uses Exercises 17.13-17.15 and proceeds as follows. By Exercise 17.13, 8(2x) is a Peano continuum; hence, by 10.8, 2°(“x) is an AR. Thus, by Exercise 17.14, n(2*‘) is an AR. Therefore, since C(X) is a retract of 2x by 10.8, we see from Exercise 17.15 that ??(C(X)) is an AR. Next, we prove that the identity map on 8(2x) and the identity map on D(C(X)) is, in each case, a uniform limit of Z-maps. For use in the proof, we devote the next two paragraphs to obtaining results that are analogous to 11.2. For any closed subset, K, of X such that K” # 8, let (cf. Exercise 14.16) a,(2X)

= {a E a(29

: I-ICI > K}

(recall 14.16). We show that ??~(2~) is a Z-set in 8(2x). Let +z > 0. Then, by 11.2, there is a continuous function fc : 2x + 2x - 2$ such that fc is within e of the identity map on 2x. Now, let f: : 22x + 22x be the maps natural induced map in 13.3. Then, it follows easily that f:18(2x) ‘i7(2x) into ??(2x) -am and that fc,‘??(zx) is within E of the identity map on n(2x). This proves that am is a Z-set in ??(2x). For any closed subset, K, of X such that K” # 0 and such that I< contains no free arc in X, let D&C(X))

= {a E B(C(X))

: na > K}.

is a Z-set in fl(C(X)); the proof is similar to the proof Then, ~K(C(X)) for ??~(2~) in the preceding paragraph. Finally, we prove that the identity maps on 8(2x) and ??(C(X)) are uniform limits of Z-maps. We use the results that we just obtained to adapt the proof of 11.3 to the present situation. Let E > 0. Replace @‘e : 2x -+ 2x in the proof of 11.3 with @: : a(ax) --+ a(2x) given by @):(a) = {C~(E, A) : A E a} for each a E n(2”); replace 2;. in the proof of 11.3 with DK, (2x). Then, by making the obvious changes in the proof of 11.3, we see that G: is a Z-map and that @z is within e of the identity map on a(2x). Also, replacing CK,(X) in the proof of 11.3 with CK, (C(X)), we see as in the proof of 11.3 that +:(??(C(X)) is a Z-map of L?(C(X)) into a(C(X)) and that G:la(C(X)) is within E of the identity map on D(C(X)). n An example of a Peano continuum, X, such that &,(C(X)) is not a Hilbert cube is in Exercise 17.16 (the example is from [8, pp. 119-1201).

140

IV. ARCS IN HYPERSPACES

For a complete see [4].

Application:

characterization

of when

Mapping

S,(C(X))

the Cantor C(X)

is the Hilbert

Fan Onto

cube,

Zx and

The term Cantor fan is the usual name given to the cone over the Cantor set. We prove the theorem in 17.10. The Cantor fan is an arcwise connected continuum, and arcwise connectedness is preserved by continuous functions (8.28 of [28, p. 1331). Thus, 17.10 implies the result in 14.9 that 2x and C(X) are arcwise connected for any continuum X. Moreover, 17.10 is significantly stronger than 14.9: There are arcwise connected continua that are not continuous images of the Cantor fan (see Exercise 17.18 for an example). 17.10 Theorem. If X is a continuum, uous images of the Cantor fan.

then 2x and C(X)

are contin-

Proof. Let K denote the Cantor set, and let F = Cone(K) (the Cantor fan). We will use the following well-known result (an especially simple proof of the result is in [28, p. 1061): (*) Every compacturn is a continuous image of K. We first prove our theorem for 2x. We obtain a map, g, of K x [0, l] onto 2x, and then we use the quotient map of K x [0, l] onto F to complete the proof. Let w be a Whitney map for 2x (13.4). Let c = {CT E &(2X) Clearly, C is closed in S,(2x). there is a continuous function, as follows: g(z, t) = [f(z)](t)

: a(1) = X}.

Hence, by 17.5, C is compact. Thus, by (*), from K onto C. Define g : K x [0, l] -+ 2”

f,

for each (z,

t)

E K x (O,l].

We show that g is continuous and that g maps K x [0, 11 onto 2x. The proof of the continuity of 9 can be done with a straightforward sequence argument that uses the continuity.of f, the continuity of each f(z), and the fact that convergence in C is uniform convergence (as noted preceding 17.1). We omit the details and point out instead that the continuity of g is the direct consequence of the following general fact: The compactopen topology for continuous functions between Hausdorff spaces is jointly continuous on compact sets [15, p. 2231. Next, we prove that g maps K x [0, l] onto 2x. Let A E Zx. Then, since X is a continuum, we see from 16.7 that there is a segment, 0, with

ORIGINAL

SOURCES

141

respect to w from A to X. Note that G E C. Thus, since f(ic) exists z, E K such that f(z,) = 0. Hence,

= C, there

dzo,O) = [f(zo)l(o) = 40) = A. Therefore, we have proved that g maps K x [0, l] onto 2x. Now, we complete the proof for 2” as follows. Recall that F = Cone(K), let w denote the vertex of F, and let x : K x [0, l] -+ F denote the quotient map: 7r(z,t)

=

(z,t), { V,

if t # 1 ift=l.

Note that g is constant on T-‘(U) since g(z, 1) = [f(z)](l) = X for all is single-valued at each point of F. Thus, (z, 1) E K x [0, 11. Hence, goV’ since 7r is a quotient map and g is continuous, g o r-l is continuous (by 3.2 of [6, p. 1231 or 3.22 of [28, p. 451). Furthermore, since g maps K x [0,l] onto 2”, clearly g 0 7r-l maps F onto 2x. This completes the proof of our theorem for 2x. To prove our theorem for C(X), let

C’ = {u ES,(C(X)): a(l) = X}; then, repeat what we did for 2x using C’ instead of C (use 16.9 where we used 16.7). W Other results about continuous images of the Cantor fan, as well as results about continuous preimages of the Cantor fan, are in [2], [23], [25], and [31, p. 1951; also, see the comment immediately following 33.11. Many of the results are also discussed in [29, pp. 81-991 (the paper that is referred to in the footnote in [29, p. 2511 will not appear; see Comment after first question in section 82 here).

Original

Sources

The result in 17.3 is 1.5 of Kelley [16]; 17.4 in the form of 17.5 is from the first paragraph of the proof of 2.7 of [16]; 17.7 and 17.8 give full generality to 1.30 and 1.32 of [29]; 17.9 is from Eberhart-Nadler-Nowell [8]; 17.10 is due to Mazurkiewicz [22] although the proof we gave is from [16, p. 251.

Exercises 17.11 Exercise. For any continuum X, S,(2x) arcwise connected continua.

and S,(C(X))

are

142

IV. ARCS IN HYPERSPACES 17.12 Exercise.

Let X be a compactum, Sk(Z)

= (0 E SW(Z)

and let 31 c 2x. Let

: (T is constant}.

Then, S&(B) is a strong deformation retract of S,,,(R); hence, S,(Z) is of the same homotopy type as 7-l. (If Z is a space and Y c 2, then Y is said to be a strong deformation retract of Z provided that there is a continuous function h : 2 x [0, l] + 2 such that h(z, 0) = z for all .z E 2, h(z, 1) E Y for all z E 2, and h(y, t) = y for all (y, t) E Y x [0, 11.) [Hint: make use of fw in the proof of 17.7.1 17.13 Exercise. If X is a Peano continuum, then 8(2x) is a Peano continuum. (We used the result in the proof of 17.9, so don’t use 17.9. A hint follows.) [Hint: Let d be a convex metric for X (10.3), and let Hj denote the Hausdorff metric for 22x induced by Hd as in 2.1. Fix Q: E D(Zx) and E > 0. Prove that {p E 8(2x) : Hi(cr,P) < E} is arcwise connected.] 17.14 Exercise. 273(2x)

For any compacturn

X, F1(8(2~))

is a retract

of

[Him: Let fw : S,(2x) + a(ax) b e as in the proof of 17.7. The key idea for the proof comes from the following obvious fact: for each o E @2”), [y: = {[f,-+)](t) : t E [O,l]}. M a k e use of Exercise 11.5. Recall 14.7.1 17.15 Exercise. Let X be a compacturn. Then, ??(C(X)) of 8(2x) if and only if C(X) is a retract of 2x.

is a retract

Remark. There are continua, X, for which C(X) is not a retract is not always a retract of 8(2”). 2” [9]. Hence, by 17.15, ??(C(X))

of

17.16 Exercise. The space S,(C(S’)) is not the Hilbert cube. (See the comments following the proof of 17.9.) [Hint: Prove that the point {S’} of ??(C(S’)) is not a Z-set in Tj(C(S’)); recall 5.2 and our discussion of Z-sets in section 9.1 17.17 Exercise. A continuum, Y, is said to be arcwise decomposable provided that Y is the union of two arcwise connected, proper subcontinua. Any nondegenerate continuum that is a continuous image of the Cantor fan is arcwise decomposable. Hence, 2x and C(X) are arcwise decomposable whenever X is a nondegenerate continuum.

18. WHEN

C(X) Is UNIQUELY

ARCWISE

CONNECTED

143

17.18 Exercise. Let Y be the Warsaw circle ((4) of Figure 20, p. 63); then, Y is an arcwise connected continuum that is not a continuous image of the Cantor fan. (See the discussion preceding 17.10.) [Hint: Use 17.17.1 Remark. If X is a continuum, then any continuum that is a continuous image of 2” or C(X) must be arcwise connected by 14.9 (and 8.28 of [28, p. 1331). By 17.10 and 17.18, the Warsaw circle is an arcwise connected continuum that is not a continuous image of 2x or C(X) for any continuum X. Results about when there are continuous surjections between continua, 2x, and/or C(X) are in Chapter XII and in [29, pp. 241-2511. The idea of relating arcwise decomposability to continuous images of the Cantor fan is due to Bellamy [2]; 17.18 is Example II of [2, p. 171.

18.

When

C(X)

Is Uniquely

Arcwise

Connected

A space, Y, is said to be uniquely arcwise connected provided that for any p, q E Y such that p # q, there is one and only one arc in Y with end points p and q. In the part of section 7 entitled Knaster’s Question, we discussedsimilarities between C(X) and Cone(X) when X is a continuum. It is easy to determine when Cone(X) is uniquely arcwise connected: The necessary and sufficient condition is that X contains no arc. This condition is also necessary for C(X) to be uniquely arcwise connected (by 5.1); however, the condition is not sufficient for C(X) to be uniquely arcwise connected (by Exercise 14.19 and the fact that there are decomposable continua that contain no arc [28, pp. 28-301). In fact, we see from 14.19 t,hat for C(X) to be uniquely arcwise connected it is necessary that X be hereditarily indecomposable. In this section we show that this condition is also sufficient: C(X) is uniquely arcwise connected if and only if X is hereditarily indecomposable. The theorem is due to Kelley (16, p. 341. We remark that hereditarily indecomposable continua were at first thought to be anomalies. In fact, when Kelley proved the theorem just mentioned, he noted that only one example of such a continuum was known (footnote 14 in [16, p. 341). N ow, however, there are many examples of hereditarily indecomposable continua (see fourth paragraph following 7.2). Furthermore, hereditarily indecomposable continua are important in, for example, continuum theory and dynamical systems. Results in this section - including lemmas - will provide us with a general understanding of how the subcontinua of an hereditarily indecomposable continuum fit together.

144

IV. ARCS IN HYPERSPACES

The main theorem is 18.8. The results of 18.8; several of them are of independent

in 18.1-18.7 facilitate interest.

the proof

18.1 Proposition. A continuum, X, is hereditarily indecomposable if and only if whenever A and B are subcontinua of X such that A cl B # 0, then A c B or B c A. Proof. Assume that X is hereditarily indecomposable. Let A and B be subcontinua of X such that AII B # 0. Then, AU B is a subcontinuum of X. Hence, A U B is an indecomposable continuum. Therefore, A c B or

B c A. Conversely, assume that X is not hereditarily indecomposable. Then there is a decomposable subcontinuum, K, of X. Hence, there are proper subcontinua, A and B, of K such that K = A U B. Clearly, A II B # 0, A $ B and B q! A. W 18.2 Proposition. Let Y be an indecomposable continuum. If A is an arc in C(Y) such that ud = Y, then Y E A. Proof.

Let h be a homeomorphism of [O,l] onto A. Let t, = inf{ t E [O, l] : Uh([O, t]) = Y)

(to exists since Uh([O, 11) = Ud = Y). Clearly, for any t > t,, Uh([O, t]) = Y; hence, using 13.3 and (2) of Exercise 11.5, we see by a simple sequence argument that (a) Uh([O,t,]) = Y. Note that if t, = 0, then h(0) = Y by (a); hence, Y E A. Thus, we may assumefor the rest of the proof that t, > 0. For each t E [0, t,], let At = Uh([O, t]) and let Bt = uh([t, to]). Note the following three properties of At and Bt: by (a), we have that (b) At U Bt = Y for each t E [0, t,]; by (3) of 11.5, we have that (c) At and Bt are subcontinua of Y for each t E [O,t,]; and, by the definition of t,, we have that (d) At#Yforanyt 0, there is a sequence, {ti}gl, in [O,to) converging to t,. By using 13.3 and (2) of converges to Bt,. Thus, since Bt, = Y for each 11.5, we seethat {Bt,}&

STRUCTURE OF ARCS IN C(X) i = 1,2,..., YEd.D

WHEN X Is HEREDITARILY...

BtO = Y. Thus, since Bt, = h(t,),

h(t,)

145

= Y. Therefore,

In relation to 18.2, see Exercise 18.9.

Structure

of Arcs in C(X) When Indecomposable

X Is Hereditarily

Assume that X is an hereditarily indecomposable continuum. Let Ao, then, by 18.1, one of A0 or Al is contained in the other or -40 n Ai = 0. Hence, the next two propositions determine the structure of all arcs in C(X).

AI E C(X);

18.3 Proposition. Let X be an hereditarily indecomposable continuum, and let Ao, Al E C(X) such that A0 c Al and AC, # Al. If A is an arc in C(X) from A0 to Ai, then d is an order arc. Proof. For each A f A - (Ao}, we let A(AoA) A from AD to A; we let d(AoAo) = {Ao}. Now, let

denote the subarc of

: A E A}.

cr = {ud(A,A)

We prove (l)-(4) below (combining (3) and (4) proves our proposition): (1) (Yis a continuum; (2) Ao,Al E cu; (3) a = A; (4) cy is an order arc. Proof of (1): Let f : A -+ C(d) be given by

f(A) = d(AoA)

for each A E A.

Obviously, f is continuous. The union map u for C(d) is continuous by (2) of Exercise 11.5. Hence, 2t o f is continuous. Clearly, u o f maps d onto a. Therefore, (1) holds. Proof of (2): Since A0 = ud(AoAa),

clearly A0 E Q. We prove that

Al E Q as follows. Let M = (A E A : Al c ud(AoA)},

ni = {A E A : Al 3 ud(AoA)}

Clearly, A1 E M; also, since A0 c Al, A0 E N. Hence, M # 0 and N # 0. Next, note that M and N are closed in d (by an easy sequence argument using the continuity of u o f in the proof of (I)). We show that d = M UN.

146

IV. ARCS IN HYPERSPACES

Let A E A. Let Z = Ud(AsA). By (3) of 11.5, Z is a subcontinuum of X; also, since A0 C Ai, Zn Ai # 0. Thus, by 18.1, -41 c Z or Z c Al. Hence, AEM (if.41 CZ)orAEN(ifZcAi). ThisprovesthatdCMUN. Therefore, A = M UN. Now, since A is connected, we seefrom the properties of M and N just verified that M fl N # 0. Hence, there exists C E Jbi fl ni. Clearly (from the definitions of M and N), Al = Ud(AaC). Therefore, AI E cy. This completes the proof of (2). Proof of (3): Since A is an arc from A0 to Al, it suffices by (1) and (2) to prove that a c A. Let Y E LY,say I’ = Ud(AeB) for some B E A. By (3) of 11.5, Y is a subcontinuum of S. Hence, Y is an indecomposable continuum. Thus, by 18.2, Y E d(AaB) (note: if B = AO then, even though 18.2 does not apply, Y E d(AoB) since Y = As). Hence, I’ E A. Therefore, we have proved that a c A. This completes the proof of (3). Proof of (4): Let, E, F E A. Then, clearly, d(AoE) C d(AoF) or d(AcF) c d(AoE). Hence, Ud(A,E) c Ud(AcF) or Ud(AsF) c Ud(AsE). This proves that cr is a nest. By (l), cy is a subcontinuum of 2.’ and, by (2), (Y is nondegenerate. Therefore, by 14.7, a is an order arc. This proves (4). By (3) and (4), d is an order arc. n Let X be an hereditarily indecomposable contin18.4 Proposition. uum, and let Ao, Al E C(S) such that AenAi = 0. If A is an arc in C(X) from 40 to Al, then A = do u di, where do is an order arc from A0 to UA and dr is an order arc from ill to UA. Proof. Let Y = ud. By (3) of Exercise 11.5, Y is a subcontinuum of X. Thus, since X is hereditarily indecomposable, Y is an indecomposable cont,inuum. Hence, by 18.2, ‘I’ E A. Also, since Aond4i = 0, clearly Y # -40 and I’ # Al. Thus, there are subarcs, de and dr , of A from A0 to Y and from ill to Jr, respectively, Since A0 c 1-, we seefrom 18.3 that do is an order arc; similarly, di is an order arc. Clearly, A = do u di . W

Uniqueness

of Arcs in C(X) When Indecomposable

X Is Hereditarily

The propositions in 18.6 and 18.7 show that when X is an hereditarily indecomposable continuum, arcs in C(X) are uniquely determined by their end points.

UNIQUENESS

OF ARCS IN C(X) WHEN

X Is HEREDITARILY..

.

147

18.5 Lemma. Let X be an hereditarily indecomposable continuum! and let ‘w be a Whitney map for C(X) (2~exists by 13.4). If A, B E C(X) such that A n B # 0 and w(A) = w(B), then A = B. n Proof. By 18.1, A C B or B C A. Therefore, since w(A) = w(B), we have by (1) of 13.1 that A = B. n 18.6 Proposition. Let X be an hereditarily indecomposable continuum, and let K, L E C(X) such that K c L and K # L. Then there is one and only one arc in C(X) from K to L. Proof. By 14.9, there is an arc, A, in C(X) from K to L. Let B be any arc in C(X) from K to L. We show that A = D. We use a Whitney map, w, for C(X) (w exists by 13.4). By 18.3, A and t3 are order arcs. Hence, it follows from the definitions in 13.1 and 14.1 that w(d) = [w(K),w(L)]

= w(B).

Now, let A E A. Then, since w(d) = w(D), there exists B E B such that w(A) = w(B). Also, A n B # 0 since A and t? are order arcs from K to L (hence, An B > K). Thus, by 18.5, A = B and, hence, A E f?. This proves that A c B. By a similar argument, 0 c A. Therefore, A = f?. n 18.7 Proposition. Let X be an hereditarily indecomposable continuum, and let Ao, Al E C(X) such that A. n Al = 0. Then there is one and only one arc in C(X) from A0 to Al. Proof. By 14.9, there is an arc, d, in C(X) from A0 to Al. Let t? be any arc in C(X) from A0 to Al. We show that A = t3. Recall our assumption that A0 n A1 = 0. Hence, A = As U dr as in 18.4 and t? = a0 U 23r as in 18.4 (with ud replaced by UB). Note that W) n W) # 0 (since Ud > A0 and UZ? > Ao). Thus, by 18.1, Ud C Ul? or UB C ud, say ud C UB. Assume first that ud = UZ?. Then, by 18.6, do = &J and di = &. Hence, A = f?. Assume next that ud # Ua. Then, since ud c Ua, we know from 14.6 that there is an order arc, C, in C(X) from ud to UB. Hence, do U C is an arc in C(X) from A0 to UB, and di U C is an arc in C(X) from Al to UB. Thus, by 18.6, do U C = f?e and di U C = BI. Hence, A C D. Therefore, since A and 23 are arcs in C(X) with the same end points, it follows immediately that A = 13.n

148

IV. ARCS IN HYPERSPACES

The Characterization We prove Kelley’s

characterization

Theorem

theorem:

18.8 Theorem. Let X be a continuum. Then, X is hereditarily composable if and only if C(X) is uniquely arcwise connected.

inde-

Proof. Assume that X is hereditarily indecomposable. Let Ao, Al E C(X) such that Ac # AI. If ACJ C A1 or Al c Ao, then there is a unique arc in C(X) from Ao to Ai by 18.6. If A0 < Al and Al @ Ao, then Ao n Al = 8 by 18.1; hence, there is a unique arc in C(X) from As to Ai by 18.7. Therefore, C(X) is uniquely arcwise connected. The converse is due to Exercise 14.19; however, in the interest of selfcontainment, we present an independent proof of the converse. Assume that X is not hereditarily indecomposable. Then there is a decomposable subcontinuum, Y, of X. Let A and B be proper subcontinua of Y such that Y = A U B. Let p E A fl B (note that p exists since Y is connected). By using 14.6 twice, we seethat there is an order arc, (Y, in C(Y) from {p} to Y such that A E cr. Similarly, there is an order arc, p, in C(Y) from {p} to Y such that B E p. Clearly, A E (Y- p; thus, Q # /I. Therefore, C(X) is not uniquely arcwise connected. n More results about C(X) when X is an hereditarily indecomposable continuum are in [16] and [29] (see [29, pp. 686-6871for a compendium).

Original

Sources

All results except 18.1 are due to Kelley [16] or can be inferred easily from ideas in proofs in [16, p. 341. In particular, 18.2 is 8.1 of [16] (a stronger result is in 1.50 of [29, p. 1021); 18.5 is 8.3 of [16]; and 18.8 is 8.4 of [16].

Exercises 18.9 Exercise. A nondegenerate continuum, X, is indecomposable if and only if X arcwise disconnects C(X). [Hint for the “only if” part: Use that a nondegenerate, indecomposable continuum is irreducible about two points [28, p. 2031.1 Remark. The result in 18.9 is 8.2 of Kelley [16]; 18.9 is also true with C(X) replaced by 2” (4.3 of [27, p. 161). For related results, see [27, pp. 15241 or [29, pp. 357-3721; in particular, a characterization Of hereditarily indecomposable continua is in 4.11 of [27] (which is 11.15 of [29] - cf. Note 1 a.t the bottom of p. 371 of [29]).

REFERENCES

149

18.10 Exercise. A continuum, X, is hereditarily indecomposable if and only if every monotone increasing sequence of arcs in C(X) is contained in an arc. (A monotone increasing sequence of sets is a sequence, {Ai},“, of sets Ai such that Ai C AZ+1 for each i = 1,2,. . ..) [Hint for the “only if” part: Make use of 18.4. (The “only if” part does not follow from 18.8: The Warsaw circle in (4) of Figure 20, p. 63 is uniquely arcwise connected and, yet, is the union of a monotone increasing sequence of arcs.)] Remark. By 18.10 and 14.9, we can apply Theorem to see that C(X) has the fixed point property whenever X indecomposable continuum. The result is due to Rogers the proof just given is from Krasinkiewicz [18, p. 1801. see 22.17 and the comment following the proof of 22.17.

16 of Young [40] is an hereditarily 133, pp. 284-2851; For more details,

18.11 Exercise. Let X and Y be nondegenerate continua such that X is hereditarily indecomposable. Then, C(X) $ Cone(Y); also, C(Y) is not embeddable in Cone(X). Remark. Regarding the first part of 18.11, it is actually the case that Cone(Y) is not even embeddable in C(X) [30, p. 2371; moreover, Y x [0, l] is not embeddable in C(X) [18, p. 1821. For related results, see [29, pp. 1491521. 18.12 Exercise. The uniquely arcwise connected continuum in Figure 25, p. 158, can not be embedded in C(Y) for any hereditarily indecomposable continuum Y. 18.13 Exercise. Let X be an hereditarily indecomposable continuum, is an hereditarily and let w be a Whitney map for C(X). Then, w-‘(t) indecomposable continuum for each t E [0, w(X)]. [Hint: 18.5 yields a natural map of X onto w-l(t).] Remark. an hereditarily

In the terminology of Chapter VIII, 18.13 says that being indecomposable continuum is a Whitney property (27.1(a)).

References 1. 2.

David P. Bellamy, Indecomposable continua with one and two composants, Fund. Math. 101 (1978), 129-134. David P. Bellamy, The cone over the Cantor set - continuous maps from both directions, Proc. Topology Conference (Emory University, Atlanta, Ga., 1970) (J. W. Rogers, Jr., ed.), 8-25.

150

IV. ARCS IN HYPERSPACES

3.

K. Borsuk and S. Mazurkiewicz, Sur Z’hyperespace d’un continu, C. B. SOC. SC. Varsovie 24 (1931), 149-152.

4.

Doug Curtis and Mark Lynch, Spaces of order arcs in hyperspaces Peano continua, Houston J. Math. 15 (1989), 517-526.

5.

D. W. Curtis and R. M. Schori, Hyperspaces which characterize homotopy type, Gen. Top. and its Appls. 6 (1976), 153-165.

6.

James Dugundji, printing,

7. 8. 9.

of

simple

Topology, Allyn and Bacon, Inc., Boston, Mass., third

1967.

Carl Eberhart and Sam B. Nadler, Jr., The dimension of certain hyperspaces, Bull. Pol. Acad. Sci. 19 (1971), 1027-1034. Carl Eberhart, Sam B. Nadler, Jr., and William 0. Nowell, Spaces of order arcs in hyperspaces, Fund. Math. 112 (1981), 111-120. Jack T. Goodykoontz, Jr., Aposyndetic J. Math. 47 (1973), 91-98.

properties

of hyperspaces, Pac.

10. Jack T. Goodykoontz, Jr., C(X) is not necessarily a retract of 2”, Proc. Amer. Math. Sot. 67 (1977), 177-178. 11. Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton University Press, Princeton, New Jersey, 1948. 12. Alejandro Illanes, Cells and cubes in hyperspaces, Fund. Math. 130 (1988), 57-65. 13. Alejandro Illanes, Monotone and open Whitney maps, Proc. Amer. Math. Sot. 98 (1986), 516-518. 14. Ott-Heinrich Keller, Die Homoiomorphie der kompakten konuexen Mengen im Hilbertschen Raum, Math. Ann. 105 (1931), 748-758. 15. John L. Kelley, General Topology, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1960. 16. J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Sot. 52 (1942), 22-36. No O-dimensional set disconnects the hyperspace of 17. J. Krasinkiewicz, a continuum, Bull. Pol. Acad. Sci. 19 (1971), 755-758. 18. J. Krasinkiewicz, On the hyperspaces of hereditarily indecomposable continua, Fund. Math. 84 (1974), 1755186. 19. J. Krasinkiewicz, Shape properties of hyperspaces, Fund. Math. 101 (1978), 79-91. 20. K. Kuratowski, Topology, Vol. II, Acad. Press, New York, N.Y., 1968. 21. A.Y. W. Lau and C. H. Voas, Connectedness of the hyperspace of closed connected subsets, Ann. Sot. Math. Pol. Series I: Comm. Math. 20 (1978)) 393-396.

REFERENCES 22. Stefan (1932), 23. Stefan Math.

151

Mazurkiewicz, SW l’hyperespace d’un continu, Fund. Math. 18 171-177. Mazurkiewicz, Sur le type c de l’hyperespace d’un continu, Fund. 20 (1933), 52-53.

24. Stefan Mazurkiewicz, SW le type de dimension de l’hyperespace d’un continu, C. R. Sot. SC. Varsovie 24 (1931), 191-192. Sur les images continues des continus, Proc. 25. Stefan Mazurkiewicz, Congress of Mathematicians of Slavic Countries (Warsaw, 1929), F. Leja, ed., 1930, 66-71. 26. M. h4. McWaters, Arcs, semigroups, and hyperspaces, Can. J. Math. 20 (1968), 1207-1210. 27. Sam B. Nadler, Jr., Arcwise accessibility in hyperspaces, Dissertationes Math. 138 (1976). 28. Sam B. Nadler, Jr., Continuum Theory, An Introduction, Monographs and Textbooks in Pure and Applied Math., Vol. 158, Marcel Dekker, Inc., New York, N.Y., 1992. 29. Sam B. Nadler, Jr., Hyperspaces of Sets, Monographs and Textbooks in Pure and Applied Math., Vol. 49, Marcel Dekker, Inc., New York, N.Y., 1978. 30. Sam B. Nadler, Jr., Locating cones and Hilbert cubes in hyperspaces, Fund. Math. 79 (1973), 233-250. 31. Sam B. Nadler, Jr., Some problems concerning hyperspaces, Topology Conference (V.P.I. and S.U.), Lecture Notes in Math. (Ed. by Raymond F. Dickman, Jr., and Peter Fletcher), Springer-Verlag, New York, Vol. 375, 1974, 190-197. 32. J. T. Rogers, Jr., Dimension of hyperspaces, Bull. Pol. Acad. Sci. 20 (1972), 177-179. 33. J. T. Rogers, Jr., The cone = hyperspace property, Can. J. Math. 24 (1972), 279-285. 34 L. Vietoris, Kontinua zweiter Ordnung, Monatshefte fiir Math. und Physik 33 (1923), 49-62. 35. A.D. Wallace, Indecomposable semigroups, Math. J. of Okayama Univ. 3 (1953), l-3. 36. L. E. Ward, Jr., intending Whitney maps, Pac. J. Math. 93 (1981), 465-469. 37. T. Wazewski, Sur un continu singulier, Fund. Math. 4 (1923), 214235. 38. Hassler

Whitney,

18 (1932).275-278.

Regular families

of curves, I, Proc. Nat. Acad. Sci.

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IV. ARCS IN HYPERSPACES

39. Hassler Whitney, Regular families of curues, Annals Math. 34 (1933), 244-270. 40. Gail S. Young, The introduction of local connectivity by change of topology, Amer. J. Math. 68 (1946), 479-494.

V. Shape and Contractibility of Hyperspaces The chapter consists of two sections. In the first section we prove that, for any continuum X, 2x and C(X) have trivial shape in the sense of Borsuk [2]. We prove this theorem in a different form at the beginning of the section; namely, we prove that, for any continuum X, 2x and C(X) are nested intersections of absolute retracts. We use this form of the theorem to obtain some basic properties of 2x and C(X). In the second section of the chapter we are concerned with contractibility. We prove two general theorems about the contractibility of 2” and C(X). We use these theorems to determine some classes of continua, X, for which 2x and C(X) are contractible. More about contractibility of hyperspaces is in Chapter XIII. The two sections of the chapter are related by the following fact: A continuum (in this case, 2x or C(X)) has trivial shape if and only if the continuum is contractible with respect to every absolute neighborhood retract.

19.

2x and C(X)

A nested intersection each i.

as Nested is an intersection,

Intersections n&Yi,

where

of ARs Yi > Yi+i for

We prove that for any continuum X, 2x and C(X) are nested intersections of ARs (absolute retracts). We then derive several important properties of 2x and C(X). 153

154

V. SHAPE AND CONTRACTIBILITYOF

HYPERSPACES

The usual approach to the results in this section is to use inverse limits. However, we have chosen the present approach so that we can obtain the results in the simplest and most accessibleway possible. We begin with the following general lemma about continua. 19.1 Lemma. tinua.

Every continuum is a nested intersection of Peano con-

Proof. Let X be a continuum. We may assumethat X is contained in the Hilbert cube Ice [17, p. 2411. Let C denote the Cantor Middle-third set in [OJ], and let VI, U2, . . . be a one-to-one indexing of the components of [0, l] - C. There is a continuous function, f, of C onto X (7.7 of [22, p. 1061). By 9.1 and 9.2, Ic*3 is an AE; hence, f can be extended to a continuous function, g, of [O,l] into Ia. Now, for each i = 1,2,. . ., let Ki = [0, I] - lJj,,uj and let Yi = g(Ki). Since Ki 3 Ki+l for each i, clearly Yi > Y,+r for each i. We prove that X = f$ZrY% and that Yi is a Peano continuum for each i. Since K, > Ki+l for each i and since fIgr K, = C, Lim Ki = C (by the converges second part of Exercise 4.16); thus, by 4.7, the sequence{Ki}fZl in 2[“l’1 to C. Hence, by 13.3, the sequence {I$}& converges in 21m to g(C). Thus, since g(C) = f(C) = X, {I$}& converges in 2’O” to X. Therefore, since Yi > Yi+r for each i, we see that X = f$?rYi (by 4.7 and the second part of Exercise 4.16). Finally, we prove that Yi is a Peano continuum for each i. Fix i. We seeeasily that K; is a finite union of closed subintervals of [O,l]. Hence, Yi is a finite union of Peano continua (by 8.17 of [22, p. 1281). We show that I< is connected. To this end, note that

thus, since yi = g(Ki)

and g(C) = f(C) = X, we see that (1) yi = 9(C u [q&+1 Vj]) = x u [u,oo,~+1g(17j)]. Since aj f~ C # 0 for each j and since g(C) = X, we seethat

(2) g(Uj) n X # 0 for each j. Since X and each g(uj) are connected, we see from (1) and (2) that 1; is connected. Therefore, since Yi is a finite union of Peano continua (as noted above), 1; is a Peano continuum (since 10.7 implies that a connected metric space that is a finite union of Peano continua is a Peano continuum). a

19.2 Theorem. sections of ARs.

For any continuum X, 2” and C(X) are nested inter-

2x, C(X)

ARE ACYCLIC

155

Proof. If X = n&Xi then, clearly, C(X) = n&C(Xi) and 2x = fl&2X*. Therefore, the theorem follows from 19.1 and 10.8. n Throughout the rest of the section we use 19.2 to derive properties of 2” and C(X). We include somegeneral background for most of the properties.

2x, C(X)

Are Acyclic

The term acyclic refers to any homology theory or cohomology theory over a coefficient group for which the theory is continuous [7]. When the coefficient group is the integers, examples of such theories are Vietoris homology, tech homology, Tech cohomology, and Alexander-KolmogoroffSpanier cohomology. 19.3 Theorem. dimensions.

For any continuum X, 2” and C(X) are acyclic in all

Proof. Every AR is contractible (Exercise 19.11). Hence, every AR is acyclic ([2, p. 861 or [8, p. 301). Therefore, the corollary follows from 19.2 (since we are assuming that our homology and cohomology theories are continuous). n

2x, C(X)

Are crANR

Our result (which is in 19.6) involves homotopy and absolute neighborhood retracts. We first give the definitions and notation that we use and prove two lemmas. We refer the reader to [2] for more information. Let Y and 2 be spaces. A continuous function from Y x [0, l] into Z is called a homotopy (a continuous function from Y x [a, b] into 2 is also called a homotopy). For a homotopy h : Y x [0, l] -+ 2 and any t E [0, 11, we let ht denote the map of Y into 2 given by ht(y) = h(y, t) for all y E Y. We say that two maps f,g : Y + Z are homotopic provided that there is a homotopy h : Y x [0, l] + 2 such that ho = f and hl = g, in which case we say that h is a homotopy joining f to g. If a map f : Y + Z is homotopic to a constant map of Y into 2, then f is called an inessential map; otherwise, f is called an essential map. A space, Y, is said to be contractible provided that the identity map of Y is inessential. We describe what it means for a space to be contractible in the following way: The space can be continuously deformed, in itself, to a point. This intuitive description is a reasonable way to envision the notion of contractibility for compact spaces;however, the description leaves a lot to be desired when a space is not compact: R’ is contracted to the point zero

156

V. SHAPE AND CONTRACTIBILITY

OF HYPERSPACES

by the homotopy h given by h(z, t) = (1 - t) . 2 for all (2, t) E R’ x [0, 11; nevertheless, ht (R1) = R’ for each t < 1. with respect to Let Y and Z be spaces. We say that Y is contractible 2 (written Y is crZ) provided that every continuous function from Y into Z is inessential (i.e., homotopic to a constant map; note that we do not require that every continuous function from Y into 2 be homotopic to the Sameconstant map). We will use the following simple lemma in the proof of 19.6. 19.4 Lemma. every space Z.

A space, Y, is contractible if and only if Y is crZ for

Proof. Assume that Y is contractible. Then there is a homotopy h : Y x [0, l] + Y joining the identity map of Y to a constant map of Y. Hence, if f is a continuous function from Y into a space Z, we see that f o h : Y x [0, l] + 2 is a homotopy joining f to a constant map of Y into 2. Therefore, Y is crZ for every space 2. The other half of the lemma is obvious. n We discussedretracts, absolute retracts, and absolute extensors in section 9. We now define the following notions. retract (written A compactum, K, is called an absolute neighborhood ANR) provided that whenever K is embedded in a metric space, Y, the embedded copy, K’, of K is a retract of some neighborhood of K’ in Y. A extensor (written ANE) compactum, K, is called an absolute neighborhood provided that whenever B is a closed subset of a metric space, M, and f : B + K is continuous, then there is a neighborhood, U, of B in M such that f can be extended to a continuous function F : U + K. The following lemma is analogous to 9.1 and is due to Borsuk. 19.5 Lemma. ANE.

A compactum, K, is an ANR if and only if K is an

Proof. The lemma follows by adjusting the proof of 9.1. Assume that K c IO3 [17, p. 2411;T is a retraction from a neighborhood, W, of K in I” onto K; B, M, f, and g are as in the proof of 9.1; let U = g-l(W); then, r o (g]U) : U + K is an extension of f to the neighborhood, U, of B in M. For the converse, let K’ c Y be as in the last part of the proof of 9.1, and simply note that, since K is an ANE, there is a neighborhood, V, of K’ in Y such that the identity map for K’ can be extended to a continuous function r : V + K’. W

2’, C(X)

ARE UNICOHERENT

157

We write Y is crANR to mean that the space Y is contractible with respect to every ANR. We are ready to prove the following theorem: 19.6 Theorem.

For any continuum X, 2x and C(X) are crANR.

Proof. We prove the corollary for 2x; the proof for C(X) is similar. Let K be an ANR, and let f : 2x -+ K be continuous. By 19.2, 2x = fP2-l Y.2 where Y,E > 1’%+r and Yi is an AR for each i. By 19.5, I< is an ANE; hence, there is a neighborhood, U, of 2x in Yr such that f can be extended to a continuous function g : U + K. Since 2x is the nested intersection of the compacta Yi and since 2x C U, there exists n such that Y,, c U (by second part of Exercise 4.16 and by 4.7). Note that since I”, is an AR, Y, is crK (by Exercise 19.11 and 19.4); also, note that g is defined on all of Y, (since I$ c V). Hence, g1Y, : Y, + I( is inessential. Thus, since 2x c Y, and 912” = f, we see that f : 2” -+ K is inessential (for if h : Y, x [0, l] + K joins g]Y, to a constant map Ic : Y, + K, then h(2X x [0, l] joins f to the constant map lc(2x). Therefore, we have proved that 2x is crK. n We note that 19.6 can not be strengthened to say that 2x and C(X) are contractible. For example, let X be the continuum in Figure 25 (top of the next page), (X consists of two harmonic fans joined at a point): Neither 2x nor C(X) is contractible (Exercise 19.12). We remark that a lot of work has been done on contractibility in connection with hyperspaces. Wojdyslawski wrote the first paper about this [30]; Kelley obtained the first general results [12, pp. 25-271. We discusscontractibility of hyperspaces in section 20 (Wojdyslawski’s result is 20.14, and Kelley’s results are 20.1 and 20.12).

2x, C(X)

Are Unicoherent

We see from 19.6 that for any continuum X, 2x and C(X) are crS1 (where S’ is the unit circle). A weaker result that is useful and more geometrically appealing is in 19.8. First, we give a definition and make some comments about the definition. A continuum, X, is said to be unicoherent provided that whenever A and B are subcontinua of X such that AU B = X, then An B is connected. The notion of unicoherence has a clear geometric interpretation: A unicoherent continuum has no hole. For example, S’ and an annulus are not unicoherent. However, the interpretation is not without flaws: The 2-sphere is unicoherent (by Theorem 2 of [18, p. 506]), and the circle-with-a-spiral

158

V. SHAPE AND CONTRACTIBILITY

X with 2x, C(X)

OF HYPERSPACES

not contractible

Figure 25

in Figure 14, p. 51 is unicoherent. Thus, unicoherent continua need not be acyclic. The proof of the following lemma uses the notion of a lift; the definition of a lift and the relevant theorem about lifts are in Exercise 19.20. 19.7

Lemma.

If a continuum is crS’, then the continuum is unicoher-

ent. Proof. Let S: = ((2, y) E S’ : y 2 0}, and let S’ = { (2, y) E 5” : y 5 0). Note th at, since Si and S! are arcs, S: and Sl are AEs (by 9.1 and 9.2). Now, assumethat Y is a continuum that is not unicoherent. Then there are subcontinua, A and B, of Y such that A u B = Y and A fl B is not connected, say A n B = E/F (section 12). Let f : A n B + S: C-ISf.

WHITNEY

LEVELS

IN C(X) ARE

CONTINUA

159

be the continuous function that is defined by letting f(E) = (1,0) and f(F) = (-1,O). s ince S: and Si are AEs, we can extend f to continuous functions gi : A + S$ and g2 : B + Sl. Let g : Y + S’ be the continuous function that is defined by letting g/A = gi and glB = g2. Note the following fact: (1) g(A II B) = {(LO), (-1,O)) = g(A) n g(B). We show that g is an essential map. Suppose that g is an inessential map. Then, by the theorem in Exercise 19.20, g has a lift cp: Y + R’. Let (with exp as in Exercise 19.20)

M = exp[cp(A)n v(B)]. Since p(A) and p(B) are intervals in R’, q(A) n p(B) is connected; hence, M is connected. However, as we now show, M = {(l,O),(-1,O)). It is obvious that

n wMB)l. (2) exp[p(An B)] c M c exp[cp(A)l Since cpis a lift for g, we seefrom (1) that (3) exp[p(A n B)] = dA n B) = {Cl, 01, NO)) and that (4) e&p(A)] n exp[dB)l = g(A) n g(B) = {(LO), (-LO)). (2)-(4), M = {(LO), (-LO)). Th us, since we have previously proved that M is connected, we have a contradiction. Therefore, g is an essential map. We have shown that if Y is a continuum that is not unicoherent, then Y is not crS’. n BY

We are now ready to prove our result about unicoherent hyperspaces. 19.8 Theorem.

For any continuum X, 2x and C(X) are unicoherent.

Proof. By 19.6, 2x and C(X) are crS’; by 14.10, 2” and C(X) are continua. Therefore, by 19.7, 2x and C(X) are unicoherent. n

Whitney

Levels

in C(X)

Are Continua

Let X be a compactum, and let 3-1c 2x. A Whitney level for ?i is any where w is some Whitney map for subset of R that is of the form w-‘(t), 3c and t E [O,w(X)]. The next theorem provides the primary motivation for the study of Whitney levels. We will discussWhitney levels extensively in Chapters VIIIX; nevertheless, we include the theorem here as an immediate illustration of the applicability of 19.8. We note that the theorem differs from previous

160

V. SHAPE AND CONTRACTIBILITY

OF HYPERSPACES

theorems in two principal ways: It is only valid for C(X) (Exercise 19.16); it is concerned with proper subsetsof C(X). 19.9 Theorem. Let X be a continuum, and let w be a Whitney map for C(X). Then, w-‘(t) is a continuum for each t E [O,w(X)J. Proof. Fix t E [O,w(X)]. Let At = w-‘([O, t]) and let f?, = w-‘([t,w(X)]). Then, by using 14.6 (or 15.3), we see that dt and & are continua (cf. Exercise 15.14). Also, as is obvious,

Hence, by 19.8, At n Bt is a continuum. Therefore, since dt we have that w-‘(t) is a continuum. w

n Bt = we1 (t),

The theorem in 19.9 says that Whitney maps for C(X) belong to a wellstudied class of maps - the classof monotone maps. A monotone map is a continuous function f : Y + 2 such that f-‘(z) is connected for each z E 2 (sometimes it is required that f-‘(z) be a continuum for each z E 2 [29]). The terminology probably comes from the fact that a continuous function from R’ into R’ is monotone in the sensejust defined if and only if the function is monotone in the usual senseof real analysis. Regarding the existence and the nonexistence of monotone Whitney maps for 2x, see [4], [ll], and section 24. A natural generalization of Whitney maps and of 19.9 is in Exercise 19.18.

2x, C(X)

Have Trivial

Shape

When only considering continua, a number of seemingly different properties are actually equivalent to being a nested intersection of ARs. One such property is that of being crANR; thus, 19.2 and 19.6 are, in reality, equivalent results. We list a few more such properties (proofs of various equivalences are in [3], [lo], and [14, pp. 237-2391): having trivial shape, being a fundamental absolute retract, being absolutely neighborhood contractible, and being a weak proximate absolute retract. The prominent role of shape theory per se leads us to reformulate 19.2 as follows: 19.10 Theorem. shape. Proof. to 19.2. a

For any continuum X, 2x and C(X)

have trivial

According to comments just made, the theorem is equivalent

ORIGINAL

Original

SOURCES

161

Sources

The result in 19.2 for C(X) is 1.5 of Krasinkiewicz [13] (cf. 1.171 of [23, p. 1751); 19.3 is from various sources depending on the type of homology or cohomology that is used and on whether 2x or C(X) is considered (notably, Kelley [12, p. 271 for 2x, who asked about C(X), and Segal [27, p. 7081 for C(X); also, Lau [19] and McWaters [20, p. 12091; for more details, see [23, pp. 176-1791); 19.6 for C(X) is 1.6 of [13], and 19.6 for 2x is 1.183 of [23, p. 1801; 19.8 is from [24, p. 4121; 19.9 is from Eberhart-Nadler [6, p. 10321; 19.10 for C(X) is 1.9 of [13], and 19.10 for 2x is 1.184 of [23, p. 1801.

Exercises 19.11 Exercise. Every AR is contractible. (We used the result proofs above; show that the result follows easily from 9.1.) 19.12 Exercise. Let X be the continuum that 2” and C(X) are not contractible.

in

in Figure 25, p. 158. Prove

19.13 Exercise. Let X be a continuum, and let K be a subcompacturn of X. Then the containment hyperspace 2,x is a nested intersection of ARs. [Hint: If Y is a Peano continuum, then 24; is an AR (see the first part of the hint for Exercise 11.6).] Remark. The result in 19.13 is also true for CK(X) is an AR by Exercise 14.22 (cf. 14.23).

since CK(X)

itself

19.14 Exercise. Let X be a continuum, and let w be a Whitney map for 2x or C(X). Then, for each t E [O,w(X)], w-‘([&w(X)]) is a nested intersection of ARs. (Compare with the next exercise.) [Hint: Use 19.1, 16.10, and Exercise 11.8.1 Remark. We see that 19.2 is a special case of 19.14 by taking in 19.14 (and recalling 13.4).

t = 0

19.15 Exercise. In relation to Exercise 19.14, consider the unit circle S’. Show that for any Whitney map, w, for 2” or C(S’), there exists t > 0 such that w-l ([O, t]) is not a nested intersection of AI&. Remark. pp. 80-831.

Results

that are related to Exercises

19.13-19.15 are in [16,

162

V. SHAPE AND CONTRACTIBILITY

OF HYPERSPACES

19.16 Exercise. Show that Whitney maps for 2” need not be monotone (i.e., the analogue of 19.9 for 2x is false) by using x’ and w in (2) of Exercise 14.25. (See section 24.) 19.17 Exercise. Let E’ be the Hawaiian Earring (Figure 26): 1’ = where Ci is the circle in R2 with center at (0,l - 2-j) and radius 2-“. Is there a continuum, x’, such that 2.’ and/or C(X) is not cry? (Compare with 19.6 - the Hawaiian Earring is one of the simplest continua that is not an ANR.) UE_,ci,

19.18 Exercise. The following notion is a natural generalization of the notion of a Whitney map. Let x’ be a compactum, and let ?l C 2.‘; a size map for ?l is a continuous function u : ?l + [0, co) that satisfies the following two conditions:

Hawaiian Earring (19.17) Figure 26

EXERCISES (1) for any A, B E ‘H such that A

c

163

B, o(A) 2 a(B);

(2) u({x}) = 0 for each {z} E 7-1. Prove the result stated below (which generalizes 19.9). If X is a continuum and D is a size map for C(X), continuum for each t E [0, a(X)].

then a-‘(t)

is a

Remark. Size maps include Whitney maps and diameter maps. The point inverses of size maps are called size levels. All the size levels for C([O, 11) are completely characterized in [25]. So far, (251 is the only paper on size maps (except for those papers about Whitney maps). Open questions about size maps are in 83.15 and 83.16. 19.19 Exercise. Let X be a continuum, let u be a size map for 2x or C(X), and let t E [O,a(X)]. Then, (T-l(t) is a continuum if and only if 0-l ([0, t]) is a continuum. 19.20 Exercise. This exercise contains the basic theorem about lifts that we used when we proved the lemma in 19.7. We state the theorem in 19.20.1 and sketch its proof; we leave the details of the proof for the reader. First, we define the notion of a lift. Let exp denote the exponential map of R1 to S’ given by exp(t) = (cos(t),sin(t)) for each t E R1. Let Y be a space, and let f : Y + S’ be continuous. A lift for f is a continuous function cp: Y + R’ such that f = exp o cp. 19.20.1. Theorem. Let Y be a compactum, and let f continuous. Then, f is inessential if and only if f has a lift.

: Y -+ S’

be

Sketch of proof. Let p denote the uniform metric for (S’)* (section 17). Verify (l)-(3) below (from which the “only if” half of the theorem follows easily). (1) Any constant map of Y into S’ has a lift. (‘4

Ifa,

c (S’JY

such that p(gi, gs) < 2, then gi has a lift if and only

if gs has a lift.

(3) If h : Y x [0, l] + S’ is a homotopy, then there exist to = 0 < t1 < t2 < . . * < t, = l(n < 00) such that p(ht,, hti+,) < 2 for each i=O,l,..., n-l. To prove the other half of the theorem, assumethat the fact that [O,l] is contractible. n

f

has a lift and use

164

V. SHAPE AND CONTRACTIBILITY

OF HYPERSPACES

Remark. The theorem in 19.20.1 remains true for any topological space Y; this was proved by Eilenberg [7, p. 681 (the proof is also in [18, pp. 426-4271).

20.

Contractible

Hyperspaces

In the preceding section we gave an example of a continuum, X, such that 2d’ and C(X) are not contractible (Figure 25, p. 158). In this section we present some basic results about when 2” and C(X) are contractible. We prove two general theorems - 20.1 and 20.12 - and we determine some classesof continua for which 2” and C(X) are contractible. We recall from section 19 that a space, Y, is said to be contractible provided that the identity map of Y is inessential (i.e., homotopic to a constant map).

The Fundamental

Theorem

We begin with what we consider to be the most fundamental theorem concerning the contractibility of hyperspaces. The theorem shows that if one of 2x and C(X) .is contractible, then so is the other. The theorem also provides a useful way to determine whether 2x or C(X) is contractible namely, if (and only if) the space of singletons, Fl(X), is contractible in 2” or C(X). (If 2 is a space and Y C 2, then Y is said to be contractible in 2 provided that there is a homotopy h : Y x [0, l] + 2 such that hc is the inclusion map of Y into 2 and hi is a constant map of Y into 2.) Theorem. For any continuum X, (l)-(4) below are equivalent: 2” is contractible; C(X) is contractible; Fl (X) is contractible in C(X); Fl(X) is contractible in 2x.

20.1

(1) (2) (3) (4)

Proof. We first prove that (1) implies (2). Assume that (1) holds. Then there is a homotopy h : 2x x [O, l] -+ 2x such that ho is the identity map of 2” and hl is a constant map. Since 2x is arcwise connected (by (1) or by 14.9), it follows easily that we can assumethat hl (A) = X for all A E 2x. We define two functions, 3 and k, on 2x x [0, 11 as follows: for each (A, t) E 2x x [0, 11,

3(A,t)={h(A,s):OQ4t} and

lc(A, t) = u3(A, t).

THE FUNDAMENTAL

THEOREM

165

Since h is continuous, it follows easily that F maps 2.’ x [0, l] into 2”-’ and that 3 is continuous. Hence, by (1) and (2) of Exercise 11.5, we see that k maps 2” x [0, l] into 2x and that k is continuous. Also, (a) k(A, 0) = U3(A, 0) = h(A, 0) = A for each A E 2x and, since h(A, 1) = X for each A E 2x, (b) k(A, 1) = W(A, 1) = X for each A E 2x. Now, temporarily, fix A E 2x. Let (YA={k(A,t):O

ifAEK if A E p-l ([0, to])

And define F : 3c x [0, l] 4 ‘R by

G(A,9, WJ,t>= A1

ifAEK if A E p-‘([O, to])

Using R and F it is easy to show that h-l ([0, to]) is a strong deformation retract of ?I!. n Using Theorem 25.19, Goodykoontz and Nadler obtained the following more definitive version of half of Theorem 25.17.

25. ADMISSIBLE

WHITNEY

MAPS

223

25.20 Theorem [15, Theorem 2.81. Let X be a continuum. Assume that there is an admissible Whitney map ,LLfor R = 2” or C(X). If 31 is contractible, then 1-1-l(to) and ,&‘([(I, to]) are contractible for each

to E P,PWl. Proof. Fix to E [0,/~(x)]. If to = 0, then p-‘(to) is homeomorphic to X and hence, by Theorem 25.17, ,u-I (to) = pL-l ([0, to]) is contractible. Thus, we assume that to > 0. We also assume that to < p(X) since if to = p(-Y), p-‘(to) = (X}. By Theorem 25.19, ~-~([0,t0]) is a retract of Ifl, thus P-’ ([O,to]) is contractible. So we only have to prove that p-‘(to) is contractible. Let G : ?f x [O,l] -+ ?t be as in the proof of Theorem 25.17. Let K = p-‘([tO, p(X)]). It follows from the properties of G that GlK x [0, 11 maps K x [0, l] into K. Hence K is contractible. By Theorem 25.19, cl-‘(to) is a retract of K. Thus p-‘(to) is contractible. n 25.21 Theorem [15, Theorem 2.91. If there is an admissible Whitney map p for 2.Y or C(X) and if X is a Peano continuum, then p-‘(to) is an absolute retract for each to such that 0 < to < p(X). Proof. 091 that 25.19, /I-’

Since X is a Peano continuum, we have by the remark in [27, P(X)]) is an absolute retract. Therefore, by Theorem is an absolute retract. W

~l([t~,

The following general result determines many properties of positive Whitney levels when the Whitney map is admissible (see Theorem 30.3, Example 30.4, Example 46.2 and Example 53.8). 25.22 Theorem. Let X be a continuum. If p is an admissible Whitney map for R = 2x or C(X), then, for each to such that 0 < to < p(X), pL-’ (to) has all those properties which are common to r-images (see [6.1 in Chapter XV] for the definition of r-image) of all hyperspaces. In particular, p-‘(to) is an arcwise connected continuum which has trivial shape (and thus it is acyclic). Proof. By Theorem 25.19, p-‘(to) is an r-image of the continuum K = /J-~([~oJ~WI>. So, in order to prove Theorem 25.22 it is enough to prove that K is an r-image of the hyperspace C(K). Define f : C(K) t K by f(d) = ud. By Exercise 11.5, f is a well defined continuous function. Define g : K + C(K) by g(A) = {A}. Then g is continuous and g is a right inverse of f. Therefore, f is an r-map from C(K) onto K. Thus p-l (to) is an r-image of C(K).

224

VII.

WHITNEY

MAPS

This completes the proof of Theorem 25.22 since the properties listed in the second part of Theorem 25.22 are known to be r-invariants and are known to be properties of all hyperspaces. n By Theorem 25.22, all positive Whitney levels have trivial shape when the Whitney map is admissible. We now show that X has trivial shape. 25.23 Theorem [15, Theorem 2.111. Let X be a continuum. If there is an admissible Whitney map p for 3t = 2” or C(X), then X has trivial shape. Proof. For each t E (0,/l-i(X)) let KCt = pP1([O,t]). By Theorem 25.19, Kt is a retract of ti. Thus, since ‘?i has trivial shape (Theorem 19.10), it follows easily that Kt has trivial shape (use [28, 2.11). Observe that Fl(X) = ll{K, : 0 < t < p(X)} Hence J’i (X) is a nested intersection of compacta having trivial shape. Thus, it follows easily that Fr(X) has trivial shape ([28, 2.11). Therefore, since Fl(X) is homeomorphic to X, _Y has trivial shape. n Recall that a cell-like map (or CE map) is a map whose point inverses all have trivial shape. So the following theorem is an immediate consequence of Theorems 25.11, 25.22 and 25.23. 25.24 Theorem [15, Theorem 2.121. Let X be a continuum. If /J is an admissible Whitney map for 2x or C(X), then p is an open CE map. In [15, Theorem 2.171, Goodykoontz and Nadler proved that, if X is a smooth dendroid, then every Whitney map for C(X) is admissible. As a partial inverse of this result we have the following. 25.25 Theorem. If X is a continuum with the property that every Whitney map for C(X) is admissible, then X is a dendroid. Proof. Let A = p-‘(to) be an arbitrary positive Whitney level for C(X). By Theorem 53.3 it is enough to prove that A is 2-connected. That is, we have to show that each map from 5”’ into A is null homotopic, where Sn is the n-sphere and 0 5 n < 2. By Theorem 25.22, p-‘(to) is arcwise connected. Then we may assumethat 1 5 n. Let 1 _