Preface
In theoretical and applied areas of mathematics we frequently deal with sets endowed with various structures. H...
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Preface
In theoretical and applied areas of mathematics we frequently deal with sets endowed with various structures. However, it may happen that the consideration of a set with a specific structure, say topological, algebraic, order, uniform, convex, et cetera is not sufficient to solve the problem posed and in that case it becomes necessary to introduce an additional structure on the set under consideration. To confirm this idea, it will do to recall the theories of topological groups, linear topological spaces, ordered topological spaces, topological spaces with measure, convex topological structures, and others. This list is not complete without adding the theory of bitopological spaces and also the theory of generalized Boolean algebras connected with certain classes of bitopological spaces. The notion of a bitopological space (X, ~-1,T2), that is, of a set X equipped with two arbitrary topologies T 1 and T2, was first formulated by J. C. Kelly in [151]. Kelly investigated nonsymmetric distance functions, the so-called quasi pseudometrics on X x X, that generate two topologies on X that, in general, are independent of each other. Previously, such nonsymmetric distance functions had been studied in [262] and in [219]. Although [151] is beyond any doubt an original and fundamental work on the theory of bitopological spaces, nevertheless it should be noted that both the notion of a bitopological space and the term itself appeared for the first time in a somewhat narrow sense in [181], [182] as an auxiliary tool used to characterize Baire spaces. For this use, the topologies T1 and ~-2 on a set X, one of which was finer than the other, were connected by certain other relations as well. Mention should be made of the viewpoint of A. A. Ivanov [137], following which a pair (X, T), where X is any set and ~- is any topological structure on X x X, is called a bitopological space in the general sense. Therefore to a bitopological space (X, ~-1,~'2) in the sense of Kelly, there corresponds a bitopological space in the general sense of the type (X, T), where ~- = T1 X 7 2 is the product topology on X x X, which in [137] is called a decomposable bitopological structure (see also [139], [141], and the interesting work [143] together with the bibliography on bitopological spaces [laS], [140], [142]). We shall adhere to Kelly's notion which at present seems to be more flexible for various usage the main objective of this monograph - developing the theory of bitopological spaces with its applications. Distance functions, uniformity, and proximity are the related notions in defining the topology and, naturally, the situation treated in [151] is by no means the
x
Preface
only way leading to a symmetric occurrence of two topologies on the same set; the investigations of quasi uniformity [190], [250] and quasi proximity [203], [124], [243] also lead to an analogous result. These topics are best covered by H. P. A. Kiinzi in [157]. Keeping in mind the symmetric generation of two topologies on a set, along with the above-mentioned cases, we can also consider ordered topological spaces [191], [208], [5], [53], [177], [178], partially ordered sets [10], and hence directed graphs [64], [65], semi-Boolean algebras [212], S-related topologies [252] and so on. On the other hand, there are many examples of nonsymmetric occurrence of two topologies on a set, particularly in general topology, analysis, and potential theory, as well as in topological convex structures (see, for example, [1], [2], [4], [7], [21], [22], [67], [260], [125], [44], [255], [172], [173]). From the above-said it follows that due to the specific properties of the considered structures two topologies are frequently generated on the same set and can be either independent of each other though symmetric by construction or closely interconnected. Certainly, the investigation of a set with two topologies, interconnected by relations of "bitopological" character, makes it possible on some occasions to obtain a combined effect, that is, to get more information than we would aquire if we considered the same set with each topology separately. If we compare all the results available in the theory of bitopological spaces from the general point of view, we shall find that in different cases two topologies on a set are not, generally speaking, interconnected by some common law that takes place for all bitopological spaces. However if, when defining a bitopological notion, the closure and interior operators are successively applied in an arbitrary initial order to the same set, then, in general, these operators will interchange in topologies as well. As a weighty argument in favour of the above reasoning, we can consider the natural bitopological space (R,a~l,aJ2), where R is the real line with the lower wl = {2~,R} U {(a,+oo) : a e R} and upper w2 = {2~,R} U { ( - o c , a) : a e R} topologies [31] playing nearly the same role in the theory of bitopological spaces as R with the natural topology co = { ~ , R } U {(a, b): a, b E R} plays in general topology and analysis as a whole. Indeed, if for an arbitrary subset A c R we take its interior in the topology c01 (respectively, co2), then the smallest closed subset, which contains this interior, is the closure of this interior in the topology co2, but not in wl (respectively, in a;1, but not in w2). Now, if for an arbitrary subset firstly we take its closure in the topology CO1 (respectively, u;2), then the largest open subset, which is contained in this closure, is the interior of this closure in the topology co2, but not in col (respectively, in u;1, but not in w2). This simple example confirms the essence of closure and interior operators, which to each subset A c R put into correspondence respectively the smallest closed set containing A and the largest open set contained in A, on the one hand, and confirms convincingly the above-mentioned interchange principle, on the other hand. In addition to our motives for studying a bitopology, that is, an ordered pair of topologies (T1, T2) on a set X, we have also derived a stimulus from G. C. L. Briimmer [50], where important problems of the same kind are referred to, in particular, hyperspaces and multivalued functions [242], [27]; function spaces [207]; H-closed,
Preface
xi
almost real-compact, nearly compact and k-compact spaces [114], [164], [231], [232]; Wallman compactifications [46], [231]; topological semifields [135]; algebraic geometry and continuous lattices [50]. It should be also noted that at present there are several hundred works dedicated to the investigation of bitopologies; most of them deal with the theory itself, but very few deal with applications. These latter papers have been published after the late 70s (see, for example, [208], [5], [53], [64], [65], [173], [50], [242], [27], [207], [114], [164], [231], [232], [46], [135], [25], [89], [90], [93]-[103], [9]). We should mention [109], [110], where J. Ewert, shows that a separable Banach space with the weak topology and the topology determined by the norm, has interesting bitopological properties, on the one hand, and gives the Baire classification of multivalued functions of topological to bitopological spaces, on the other hand, and [249] since according to its author J. Swart the axiomatic topological characterization of Hilbert spaces is due to a large extent to the bitopological analogue of the notion of an open cover from [113]. In the above context we can also recall [258], where the term "consistent" equivalent to "bitopological Hausdorff" is one of the key notions, and [173], where the "bitopological boundary" is essentially used for establishing the minimum principle for finely hyperharmonic functions. The theory of bitopological spaces and its applications owe much to J. M. Aarts [3], [4]; D. Adnadjevid [5], [6]; S. P. Arya [16]-[19]; B. Banaschewski [23]-[251; T. Birsan [301-[33]; G. C. L. Briimmer [46]-[52]; A. Csgszgr [69], [70]; M. C. Datta [72]-[74]; J. Dei~k [76]-[78]; D. Doitchinov [80], [81]; P. Fletcher [112]-[117]; M. Jelid [144]-[147]; Y. W. Kim [152], [153]; H. P. a. Kiinzi [154]-[159]; E. P. Lane [165][167]; M. Mrgevid [3], [4], [183]-[188]; M. G. Murdeshwar [189]; S. a. Naimpally [189], [194], [195]; C. W. Patty [202]; W. J. Pervin [203]-[205]; H. A. Priestley [208]; I. L. Reilly [213]-[218]; S. Romaguera [221]-[225]; M. J. Saegrove [228]; S. Salbany [229]-[233]; A. R. Singal [235]-[238], [240]; M. K. Singal [235], [237]-[239]; J. Swart [248], [249], and to many other authors, who are not listed here and to whom we offer our apologies. This monograph is a versatile introduction to the theory of bitopological spaces and its applications. It considers the topics of bitopology that were studied perfunctorily or not studied at all and presents original results and examples which, we dare think, will stimulate the reader to further research. The monograph consists of eight chapters, of which Chapters III, IV, V, VI, VII form the core because they contain the basic results related to the abovementioned topics. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, theory of ordered topological spaces, and graph theory. Moreover, a high level of modern knowledge of bitopological spaces theory has made it possible to introduce and study an algebra of new type, the corresponding representation of which brings one to the special class of bitopological spaces.
xii
Preface
To conclude the preface, we would like to note that we firmly believe that from the standpoint of applications the theory of bitopological spaces has no less promising prospects than the theory of topological spaces. The areas of such applications are, in our opinion, the theories of linear topological spaces and topological groups, algebraic and differential topologies, the homotopy theory, not to mention other fundamental areas of modern mathematics such as geometry, analysis, mathematical logic, the potential theory, the probability theory and many other areas, including those of applied nature. In particular, the study of strong and weak topologies in analysis, the initial and the Alexandrov topologies on a manifold in the global Lorentzian geometry, cohomologies of spaces with two topologies, and the theory of foliations seems very promising for future research.
CHAPTER 0
Preliminaries Besides being auxiliary, this chapter also contains the internal characterization of pairwise completely regular bitopological spaces. In Section 0.1, along with the symbols and notations, we give a survey of the basic concepts and results from the theory of bitopological spaces to be used in our further investigation. In particular, we recall various kinds of pairwise separation axioms and their interrelations established by J. C. Kelly [151]; E. P. Lane [166]; M. G. Murdeshwar and S. A. Naimpally [189]; J. Swart [248]; I. L. Reilly [215], [217]; Y. W. Kim [152]; T. Birsan [31]; A. R. Singal [236]; M. K. Singal and a. R. Singal [238]; C. W. Patty [202]; D. N. Misra and K. K. Dube [180]; M. J. Saegrove [228]; W. J. Pervin and H. Anton [205], and others. Since the study of relations between the theory of bitopological spaces and some other branches of mathematics in Chapter VII demands special knowledge of bitopologies, we recall the appropriate notions of pairwise compactness, pairwise local compactness, pairwise paracompactness, pairwise local LindelSf property, and pairwise paraLindelSf property. These notions were formulated for the first time by P. Fletcher, H. B. Hoyle, and C. W. Patty [113]; J. Swart [248]; M. C. Datta [72]; M. Mrgevid [183], [184]; R. A. Stoltenberg [245]; I. L. Reilly [214]; T. G. Raghavan and I. L. Reilly [209]; T. Birsan [30]. We also present bitopological versions of connectedness and similar type properties, the study of which was initiated by W. J. Pervin [204]; H. Dasgupta and B. K. Lahri [71]; J. Swart [248], and C. Amihg~esei [12]. Consideration is given to the bitopological notions of continuous, open, closed, and homeomorphic maps introduced by J. Swart [248] and A. R. Singal [2361. In the topological case the complete regularity in internal terms, that is, without using the notion of a function, was characterized by O. Frink [119], E. F. Steiner [244], and V. I. Zaicev [264]. Their modifications for bitopological spaces were studied by Saegrove, who used the generalization of Steiner's method, and by us with the aid of the generalized method of Frink and Zaicev. 0.1. S y m b o l s a n d N o t a t i o n s . Basic C o n c e p t s of B i t o p o l o g y Throughout the book, along with the generally accepted symbols, we use our own notations and those from [68] and [200]. Sets are usually denoted with italic capitals A, B , . . . and elements of sets with lower case italic a, b, . . . . Sets whose elements are sets are called families of sets, while their elements are called members. Families of sets and classes of functions,
2
0. P r e l i m i n a r i e s
with rare exceptions, are denoted with one or two script letters A, B , . . . , followed by one italic capital in brackets for families of sets signifying spaces, and two italic capitals in brackets, in the case of classes of functions, denoting the corresponding spaces. The empty set is denoted by ;~, while the symbols N, Z, Q and R are respectively used for the sets of all natural numbers (excluding zero), of all integers, of all rational numbers, and of all real numbers. We also use oc to indicate "an infinite number". Other standard symbols are defined for each n E N as follows: n - 1, k means that n E { 1 , 2 , . . . , k } and n - 1, oc means that n E { 1 , 2 , . . . }. The closed (open) real segment joining a and b, denoted by [a, b] ((a, b)), is the set { x c R " a_<x_ 0}. This definition immediately implies that a n / - z e r o set is/-closed. Now we can proceed to prove our main result, that is, Theorem 0.2.5. Necessity. Let (X, TI, ~'2) be a p-completely regular BS in the sense of (14) of Definition 0.1.6. Also, let Z - {Z1,Z2}, where Zi is the family of all i-sero sets. By [166, Proposition 2.9], Z - {Zx, ~-2} is a d-closed base of (X, T1, T2). Let us show that conditions (1) and (2) of Definition 0.2.4 are satisfied. (1) Let x 9 X be any point and g(x) 9 co Zj be its any neighborhood. Since (X, Wl, T2) is p-completely regular, the point x is (i,j)-completely separated from the j-closed set X \ U(x), that is, there exists an (i, j)-l.u.s.c, function
{x C X "
f " (X, TI,~'2) --, (I,a/) such that f ( x ) -
0 and ( f ( X \ U ( x ) ) - 1.
Assume that A - f - l ( 0 ) . Clearly, A 9 Z~ and x 9 A c U(x). (2) Let A 9 Z1, B 9 Z2 and A n B - ~. Then there exist a (1,2)-l.u.s.c. function r _> 0 and a (2,1)-l.u.s.c. function ~ _> 0 such that A - r B - ~-1(0). By [166, Proposition 2.8], there exists a (1,2)-l.u.s.c. function h" (X, T1, w2) ~ (I, co') such that h(A) - 0 and h(B) - 1. Let 0 hi(x)
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22
I. Different Families of Sets in Bitopological Spaces
Pro@ (1) Using Theorem 2 from [161, p. 78], we obtain {x E X :
T2 el A-~ 1-Bd(X, x) } = 7"1 c17"1 int 7"2 el A.
Thus, it remains to use Proposition 1.1.19. (2) It is clear that { X C A"
A c (1, 9~)-.~:)(X, x) } - A \ 7"1 cl 7"1 int 7"2 cl A c 7"2 cl A \ 7"1 int 7"2 cl A
and 7"1 int(T2 el A \ T1 int 7"2cl A) = n i n t 7"2cl A A (X \ 7"1 el 7"1 int 7"2cl A) = ;~. Therefore 7"2clA \ 7"1 int T2 clA E c07"2 A 1-Bd(X) C (1, 2)-A/D(X).
K]
In particular, if A is (1, 2)-nowhere dense at each of its points, then A = A \ 7"1 cl 7"1 int 7"2cl A so that 7"1 int 7"2 clA = 2~ A e (1, 2)-AfD(X). D e f i n i t i o n 1.1.21. A subset A of a BS (X, 7"1,7"2) is termed (i, j)-somewhere dense in X if A-~ (i, j)-AfD(X), that is, if T~ int 7"j cl A ~ ~. The families of all (i,j)-somewhere dense subsets of X are denoted by Clearly, (i, j ) - S D ( X ) = 2 X \ (i,j)-N'D(X) and hence for the natural BS (R, aJl,W2), we have aJi \ {~} c i-D(R) = 2 R \ i-AfD(R) = i - $ D ( R ) . It is easy to ascertain that the inclusions
(i, j ) - S D ( X ) .
(1,2)-SD(X) c 2-SD(X) A A 1 - $ D ( X ) C (2, 1)-SD(X) hold for a BS (X, T1 < 7"2). In order to use category notions to study bitopological concepts of Baire spaces, we must relate them to the bitopology on a set. D e f i n i t i o n 1.1.22.
A subset A of a BS (X, T1,72) is of (i,j)-first category oo
(also called (i,j)-meager, (i,j)-exhaustible) in X if A -
[.J An, where An e n=l
(i,j)-AfD(X) for every n = 1, oc and A is of (i,j)-second category (also called (i, j)-nonmeager, (i, j)-inexhaustible) in X if it is not of (i, j)-first category in X. A subset A of X is of (i, j)-first category if A is of (i, j)-first category in itself and A is of (i, j)-second category if it is of (i, j)-second category in itself. The families of all sets of (i, j)-first ((i, j)-second) categories in X are denoted by (i, j)- Catg~ (X) ((i, j)- Gatgii (X)), while the statement X c (i, j)- Catg I (X) (X ~ (i, j)-Catgii (X)) is abbreviated to X is of (i, j)-Catg I (X is of (i, j)-Catg II). It is clear that
(i, j)-AfD(X) c (i, j)-datg~ (X) = 2 X \ (i, j)-Catgii (X). R e m a r k 1.1.23. Let T1 = ~ and T2 be respectively the natural and the discrete topology on R. Then n e N implies {n} e (1,2)-AfD(R) and hence N e ( 1 , 2 ) - d a t g i ( R ). On the other hand, if 7~ and T~ are the corresponding
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R0. Indeed, let U(x) c T1 be any neighborhood. Then there is a set Un from the 1-countable base 0"1, 0"2,... such t h a t z c Un c U(x). It is obvious t h a t Un /: Vk for each k since the contrary implies z g F \ U vk = A. k=l
This means that IF a U~] > R0. Since A differs from F merely in a countable set, it follows that [ANU~ I > R0. But A N U ~ c A N N ( x ) so that I A N U ( x ) [ > b~0 and so A c A ~ Therefore A c AId and thus A c co ~-2 n p-7927(X) - (2, 1)-7)(X). Now, we have A c A ~ c A d so t h a t T1 C1 A C T1 C1 A~ -
Therefore A ~
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cl A. Finally, (x)
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C o r o l l a r y 1.4.18. Under the hypotheses of Theorem 1.4.17, for any uncountable set B C X there is an uncountable subset A C B such that A c p-T)Z(X).
Proof. Let B c X be any uncountable set. Then the result follows directly from the proof of Theorem 1.4.17 omitting the remark that A is 2-closed. D At the end of this section we shall consider three operators on 2 x , which are used to characterize degrees of nearness of the four boundaries of a set, the S-, C- and N-relations in Chapter II and interrelations of dimension functions in Chapter III. D e f i n i t i o n 1.4.19. For a BS (X, T1 < ~-2) the indicators of nearness of the boundaries are the following three operators: n l , n2, n : 2 X --~ (2, 1)-12C(X), defined as follows: nl
(A) = T1 cl A \ 7-2 cl A, n2(A) = ~-2 int A \ T1 int A
and n(A) - n l ( A ) u n2(A) for each set A c 2 X. It is obvious t h a t n~(A) - n j ( X \ A) so t h a t n(A) - n ( X \ A) for each set A c 2 X, the restrictions nllCOWl
-- n2171
--
n
71neoY 1 z
and, therefore, nlw I - n l ,
n[co~-i - n 2 .
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1.5. Relative Properties
51
If A 9 (72 \ 7"1)n CO 7-1, then n ( A ) -- n 2 ( A ) - A N T1 c l ( X \ A) - T1 c1A AT1 c l ( X \ A) - 1 - F r A -
=T2clANTlcl(X\A)-(2,1)-F'rA, If A
r (~-2 \ 71) N (co 7-2 \ co 7-1),
(1, 2)- Fr A - 2- F r A - 2~.
then
n l ( A ) - 71 c I A n ( X \ A) - 7-1 c l A n
T2 c l ( X \ A) - (1, 2)- E r A ,
n 2 ( A ) - A n T1 c l ( X \ A) - T2 C1A n T1 c l ( X \ A) - (2, 1)- Fr A,
n(A)-(1,2)-FrA
U (2,1)-FrA-(1,2)-FrAA(2,1)-FrA-I-FrA,
2-FrA-2~.
(3) The proof consists of elementary calculations taking into account the fact that the equality nl(A) - n2(X \ A) is fulfilled for each set A 9 2 x. (4) If A 9 p-TPZ(X), then A~ c A d - (A1d \ A d) U A f so that A; c (A d \ A d) - (A U A1d) \ (A-U A d) - nl (A). Conversely, A~ c nl(A) implies A~ c A1d \ A~ c A1d and so A 9 p-7:)Z(X).
[]
1.5. R e l a t i v e P r o p e r t i e s Relative properties, that is, to say, such properties of subsets of subspaces of TS's that are preserved from spaces to subspaces, from subspaces to spaces or in both directions, were investigated in various published works on general topology. Naturally, there arises a question how widespread relative properties are in the theory of BS's because these properties will play an important role in our further investigations. D e f i n i t i o n 1.5.1. Let (X, T1,7-2) be a BS and (Y, r~, r~) be a BsS of X. If A c Y, then (1) A c (i, j)-D(Y) if one of the following equivalent conditions is satisfied:
Y-T~clTjclANY
~
Y c r~ clw] clA ~
Y - r~'clw] clA.
(2) A c (i, j)-Bd(Y) if one of the following equivalent conditions is satisfied: Y - 7-~cl(Y \ ~-~int A) N Y Y C 7-i cl(Y \ 7-] int A) < :, < :- Y - ~-: cl(Y \ T] int A) ~
~-~int r~ int A - 2~.
(3) A c (i, j)-A/'TP(Y) if r] clA c i-13d(Y)so that if one of the following equivalent conditions is satisfied: Y - Ti cl(Y \ ~-] el A) N Y +---> Y C ~-i cl(Y \ ~-] el A) < > Y - r~ cl(Y \ 7-~ cl A) , T~ int r] cl A - 2~. (4) A c (i, j)-SlP(Y) if one of the following equivalent conditions is satisfied: Y r ~-i cl(Y \ Tj cl A) n Y , Y \ T~cl(Y \ T] cl A) / 2~ ,' < ;-Y # T/cl(Y \ T ] c l A ) ~
T~intT~clA =/= 2~.
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1.5. Relative Properties
53
C o r o l l a r y 1.5.4. Let (X, 7.1 ( 7 " 2 ) be a BS, (7.2clY, 7.~,7.~)and (Y, 7.~',7.~') be BsS's of X , and A c 7.2 cl Y. Then the following conditions are satisfied: A c (2, 1)4VD(~2 d Y) --*, A n Y c (2, 1)-ArD(Y) so that
A n Y c (2, 1)-$/P(Y) ---> A c (2, 1)-S:D(% cl Y) and thus A E (2,1)-Catgi(7.2clY) ~
A N Y c (2,1)-Catgi(Y),
A N Y E (2,1)-Catgii(Y)
A c (2,1)-Catgii(7.2clY).
~
Proof. It suffices to show only the validity of the first implication. By Proposition 1.5.3, Ac(2,1)-AfZ~(7.2clY),z--->7.; c l A c % el (%elY \ 7.~ c l A ) - % el ( % e l y \
7.1
clA),
while (2) of Lemma 0.2.1 gives 7.; cl A c 7.2 cl (7.2 cl Y \ % el 7.1 el A) C 7.2 cl(Y \
7.1
cl A).
Hence 7.; cl(A n Y) c % cl(Y \ 7.; cl A) c % cl (Y \ 7.; cl(A n Y)). But A n Y c Y c T2 cl Y ~
T 1" c l ( A n Y ) - -
T 1' c l ( A N Y )
NY
and thus T;' cl(A N Y) C 7.2 cl (Y \ T; cl(A N Y)) - T2 cl (Y \ T;' cl(A N Y)). Hence, again applying Proposition 1.5.3, we find that A n Y c (2, 1)-A/T)(Y).
D
By analogy with the second part of Theorem 1.1.3, we obtain the sufficient conditions for the relative (i, j)-nowhere densities. P r o p o s i t i o n 1.5.5. Let (X, 7-1,72) be a BS and (Y, 7.~, 7.~) be a BsS of X, where Y c j-7?(X) and A c Y . If for every set V c 7.~ \ {~} there exists a set V e vj \ {~} such that V n A = 2J, then A e (i,j)-A/Z)(Y). For every set U' c T~\{2~} there exists a set U E Ti\{2~} such that U N Y = U'. Hence, by assumption, there is a set V ~ rj \ {2~} such that V c U and V N A = ~. But r j c l Y = X implies that V N Y = V' C r j \ { ~ } and so it remains for us to use the second part of Theorem 1.1.3. D Pro@
7-1,7-2) be a BS and (Y, 7.~, 7.~) be a BsS of X , where Then the following statements hold:
T h e o r e m 1.5.6. Let (X, Y c 7.i and A c X .
A ~ ( i , j ) - A f Z ) ( X ) --->, A n Y E (i,j)-Af:D(Y) so that A n Y c ( i , j ) - S Z ) ( Y ) ----> A E ( i , j ) - S T P ( X ) and thus A c ( i , j ) - d a t g i ( X ) --->, A N Y C (i,j)-Catg~(Y), A n Y C (i,j)-Catg~i(Y) ~
A c (i,j)-Catgii(X).
(7-1int A ~ 2~ ~
,,.
;.
7-2 int A ~= 2~ for every subset A c X ) .
Thus, by Proposition 1.1.11, for the equivalence (1) ,z--> (2) it is enough to show that (1) ,z--->, (7-1 int A r ~ ~ 7-2 int A ~ ~ for every subset A c X). We assume that 7-1S7-2, that is, 7-1 \ {2~} is a pseudobase for 7-2, and A c X is any subset. If 7-1 int A ~= z , then by (1) of Definition 2.1.1, 7-2 int 7-1 int A # ~, and hence 7-2 int A # ~. When 7-2 int A # z , by (2) of Definition 2.1.1, there exists a set V E 7-1 \ {Z } such that V c 7-2 int A. Thus 7-1 int 7-2 int A # 2~ so that 7-1 int A # ~. On the other hand, let U e 7-1 \ {2~}. Then 7-2 int U :/= 2~. If V e 7-2 \ {2~}, we have ~ r V = 7-1 int U c U and, consequently, 7-1 \ {2~} is a pseudobase for 7-2. Therefore (1) .z--> (2). The implication (1) ~ (3) is exactly Proposition a.a from [252], (2) is obvious. (3) ,z----5, (4) is an immediate consequence of Theorem 1.3.12. D It is likewise easy to see that if 7-1 = a~ is the natural topology on R and 7-2 is the discrete topology on R, then 7-1 is not S-related to 7-2. In the sequel it will be assumed that (7-1S7-2 A 7-1 C 7-2) ~ 7-1 < s 7-2 and the corresponding BS will be denoted by (X, 7-1 < s 7-2). Furthermore, Example 2.1.4 shows that for a BS (X, 7-1 < s 7-2) the equality 7-1 = 7-2 does not hold in general. C o r o l l a r y 2.1.6. The following conditions are satisfied for a BS (X, 7-1S7-2): 9
(1) X[ - X~ and thus if either of the topologies
7-1 and 7-2 is discrete or antidiscrete, then 7-1 = 7-2. (2) (X, 7-1,7-2) is d-quasi regular if and only if (X, rl, 7-2) is p-quasi regular. (3) If (Y, 7-~,7-~) is a BsS of X , where Z 9 (7-1 N 7-2)[-J 1 - ~ ) ( X ) 2-~:)(X), then 7-~S7-~.
Proof. The proof of (1) is straightforward. (2) We begin by assuming that (X, 7-1,7-2) is a 1-quasi regular and 2-quasi regular BS and U c 7-~\ {~}. By t h e / - q u a s i regularity of (X, 7-1,7-2), there is a set V E 7-~\ {~} such that 7-~cl V C U. But 7-1S7-2 implies that 7-j int V = W r ~ and by the j-quasi regularity of (X, 7-1,7-2), there exists a set E c rj \ {~} such that rj c l E C W. Let O = 7-~int E. Then O r ~ and 7-j cl O c U so that (X, rl, 7-2) is (i, j)-quasi regular. Conversely, if (X, 7-1,7-2) is (1,2)-quasi regular and (2, 1)-quasi regular, then for a set U c r~ \ {~}, there is a set V c 7-~ \ {~} such that 7-j el V c U. For the set r j i n t V = W r ~, there is a set E c rj \ {~} such that r ~ c l E c W. Let O = 7-i int E. Then O r ~ and ri cl O c U so that (X, 7-1,7-2) is/-quasi regular. (3) First we assume that Y E 7-1 N 7-2 and U E 7-~ in (Y, 7-~,7-~). Then U E 7i and by (4) of Theorem 2.1.5, U c (j, i ) - $ O ( X ) . Hence, according to (1) of Proposition 1.5.26, U c (j, i ) - S O ( Y ) so that 7-~ c (j, i ) - S O ( Y ) and, again applying (4) of Theorem 2.1.5, we obtain 7-~$7-~. On the other hand, let Y c 1-Z)(X) = 2-79(X) and A c Y, r~ int A/= ~. Then, there is a set U c 7-i such that U ~ Y = 7-~int A. But 7-1S7"2 implies 7-j int U = V r 2~
66
II. Different Relations Between Two Topologies . . .
and, therefore, V Cq Y = V' # ;~ as Y 9 1-D(X) = 2-D(X). Thus Tj int A # since V' C r" int A. D C o r o l l a r y 2.1.7. The conditions below are equivalent for a BS (X, rl, r2): (1) rl < s v2. (2) 1-13d(X) c 2-13d(X) so that 1-D(X) C 2-D(X). (3) r2 int r2 cl A = re int T1 cl A so that r2 cl r2 int A = r9 cl r l i n t A
for every subset A c X . (4) r2 c (1,2)-SO(X) so that cot2 c (1,2)-SC(X). (5) n(A) 9 2-Bd(X) for each set A 9 2x.
Furthermore, the next statement holds: (6) If (Y, 7-;, r~) is a BsS of X and Y 9 r2, then T 1 S T 2 -----5, T 1
Proof. The equivalences (1) (2) ,e--->, (3) (4) are immediate consequences of the corresponding equivalences of Theorem 2.1.5. (1) ------5, (5). If 7"1S7"2, then by (b) of 4.A.2 in [173], nl(A) 9 2-A/'D(X) and n2(A) 9 2-A/'D(X) for each set A c 2 x. Therefore n l ( A ) c 2-Bd(X) and n2(A) E 2-Bd(X) for each set A E 2x, and thus n(A) c 2-Bd(X) for each set A E 2x. (5) ---> (4). Clearly, if n(A) 9 2-Bd(X) for each set A 9 2x, then n2(A) 9 2-Bd(X) for each set A 9 2 x. Hence ~-2 int(r2 int A \ r l i n t A) = ;~ for each set A c 2 x so that r2 int A c r2 cl r l i n t A for each set A c 2X, that is ~-2 C (1, 2)-$(.9(X). (6) Let A c Y and r ~ i n t A ~= ~. Then r 2 i n t A # ~ and, by (2) above, r l i n t A ~: 2~. But r l i n t A c r~ int A and so r; Sr~. [5] Further, following (1) of Corollary 1.3.15 and (4) of Theorem 2.1.5, for a BS have
( X , T1, T2), w e
rl < s r2 ~
(rl c r2 c (1,2)-,,gO(X)c ( 2 , 1 ) - $ 0 ( X ) ) .
Note that in contrast to (3) of Corollary 2.1.6 and (6) of Corollary 2.1.7 the S-relation is not, generally speaking, hereditary with respect to/-closed subsets. E x a m p l e 2.1.8. Let X = {a,b,c,d}, 71 = {Z, {a, b}, {a, b, c}, X } and 72 = { z , { a , b } , { a , b , c } , { a , b , d } , X } . Clearly, rl < s r2, but if F = {c,d} e cot1 c co 72, then r~ is not S-related to v~ for the BS (F, v~, v~). This fact leads to the following notion to be used in Chapter IV.
b-
"~-J
=.~
rat]
9-
~@
9~
~
o ~
9,-,
o,~
~
~
o
~
~
,~
II
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~
,
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~
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@
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-
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u~
I
~
X is of (1, 2)-Catg II e---> X is of (2, 1)- CatgII ---> 1- G~(X) N 2-D(X) = 1- Ga(X) N 1-D(X) c 2-Ga(X)N1-D(X) = 2 - G a ( X ) N 2 - D ( X ) C 2-Catgut(X)= 1-Catg~i(X ) = (1,2)-Catgix(X) = (2, 1)-CatgI~(X ).
Proof. First assume that A c X is any subset. Then by (3) of Theorem 2.1.5, we have ~-i int rj cl(i, j)- Fr d = ~-i int rj cl (ri cl A a rj cl(X \ d)) c c T~int rj cl ~-~cl A a r~ int rj cl(X \ A) c ~-~cl A N T~cl(X \ A) = i- Fr A. Similarly, riint rj cl(j, i)- Fr A c i- Fr A. If A E T~ U co r~, then by Theorem 3 from [161, p. 74], i-FrA E i-A/'D(X). Hence by (1) of Theorem 2.1.10 and the inclusions above, riint rj cl(k, 1)- Fr A e 1-HD(X) = 2-A/'D(X) = (1, 2)-HD(X) = (2, 1)-A/'D(X)
2.1. The S-Relation
71
and so 7-i int 7-j cl 7-i int 7-j cl(k, l)- Fr A = 2~, where k, l E {1, 2}, k ~ 1. Thus by (1) of Lemma 0.2.1,
(i,j)-Fr A c 1-AF/?(X) = 2-AF/9(X) = (1, 2)-A/'/9(X) = (2, 1)-AF/9(X) for each set A c 7-1 d co 7-1 U 7-2 U co 7-2. (1) It suffices to show that n l ( A ) E 1-Af~D(X) since n2(A) = n l ( X \ A). Obviously, T 1 int 7-1 cl
(7-1 cl A \
7-2 cl
A) c
and, by (3) of Theorem 2.1.5 for
7-1 int 7-1 cl
T 1 C 7"2,
7-1 int
A c~ (X \
T1
cl 7-1 int 7-2 cl A)
we have
rl cl A C 7-2 cl A.
Hence T1
int
7-1 cl
A c 7-1 cl 7"1 int 7-2 cl A
so that T1 int T1 cl A c~ (X \ q cl 7-1 int r2 cl A) = 2~ and thus n 1 ( A ) --- 7-1 cl
A \ 7-2 cl A ~ 1-A/'Z)(X).
The proof of (2) follows directly from Proposition 1.1.26, (2) of Theorem 2.1.5, and (1) of Theorem 2.1.10. [-] R e m a r k 2.1.14. According to [252] let us treat a TS (Y,'7) as the image of a TS (X, r2) under a continuous function f with the property: for every set U c 7-2 \ {2~}, there exists a set V c 7 \ {2~} such that f - l ( v ) C U. Clearly, the family 7-1 = f - l ( ~ ) = { f - l ( p ) : V E ~ } is a t o p o l o g y o n X c o a r s e r t h a n 7-2. I f w e denote this relation between the topologies 7-1 and 7-2 on X by 7-1f72, then 7-1fr2 is a stronger connection between the topologies than the <s-relation. Hence all results obtained for the <s-relation also hold for the f-relation. Furthermore, following [201], a condensation is a one-to-one and continuous function f : (X, 7-2) ~ (Y, "y) such that I ( X ) = Y. It is clear that in this case f : (X, 7-1) --+ (Y, 3') is a homeomorphism and, conversely, for any BS (X, 7-1 < ~-2) the identity function f : (X, 7-2) ---' (X, 7-1) is a condensation. Theorem
2.1.15. If f : (X, 7-2) ---' (Y, 7) is a condensation and 7-1 = f - l ( . y ) ,
then 7-1 <s 7-2 ,z----5, TlfT-2 ~
f is feebly open
(see Definition 5.1.37). Proof. By Remark 2.1.14 for the first equivalence, it suffices to prove that T1 < s T2 implies TlfT2. Let U c T2 \ {2~} be any set. Then T1 int U c 7-1 \ {2~} and, hence, there is a set V c 7 \ {2~} such that f - l ( V ) = 7-1 int U C U, that is, 7-1f7-2. Now, let 7-1f7-2 and U ~ 7-2 \ {2~} be any set. Then there is a set V E 3' \ {2~} such that f - l ( V ) c U. Therefore f(f-l(v))
= V c f(U),
that is Tint f ( U ) r Z
72
II. Different Relations Between Two Topologies ...
and thus f is feebly open. Conversely, let f be feebly open and U c 7"2 \ {~} be any set. Then the set Tint f(U) - V e T \ {2~} satisfies the inclusion f - l ( v ) C U and so 7"1f7"2. E] C o r o l l a r y 2.1.16. If f " (X, T2) ~ (Y, T) is a feebly open condensation, (Y, T) is quasi regular, 7.1 - f - l ( T ) and (X, 7.1,7.2) is 1-quasi regular, then (X, 7.1,7"2) is d-quasi regular and p-quasi regular.
Proof. It is obvious that 7"1f7"2 and by Proposition 3.5 from [252], (X, T1,7-2) is 2-quasi regular. Therefore (X, 7.1,7"2) is d-quasi regular and by (2) of Corollary 2.1.6, (X, T1,7"2) is p-quasi regular since T1 < S 7"2. [-] By the above reasoning, if for a BS (X, 7"1 < 7"2) the identity function (i.e., the inclusion function) j ' ( X , 7.2) ~ (X, 7 - 1 ) i s feebly open and (X, 7.1,7.2)is 1-quasi regular, then (X, 71, ~-2) is d-quasi regular and p-quasi regular. 2.2. T h e C - R e l a t i o n Let (X, 7.1,7.2) be a BS. Then it is obvious that z2 c 7"1 ~ 7"1 cl A c 7"2cl A for every subset A c X. In [258] a topology 7"1 is chosen among different subfamilies of 2 x whose elements satisfy inclusions of the type given on the right-hand side of the above equivalence. This section continues the study begun in [258], while the Section 2.3 will deal with the same inclusion for a topology 7"2. D e f i n i t i o n 2.2.1. A topology T1 is coupled to a topology 7.2 on a set X (briefly, 7.1C7.2) if 7.1 cl U C T2 cl U for every set U c 7.1 [258]. From this definition we immediately find that if 7.1 - co 7"1, then 7"1 is coupled to every topology on X so that the antidiscrete topology on X as well as the discrete topology on X is coupled to every topology on X. R e m a r k 2.2.2. By [258], if T1 is coupled to z2 on X, then T1 is coupled to every topology on X smaller than 7.2. But it is possible for a topology to be coupled to a strictly larger topology and in that case the coupling is mutual. For example, the antidiscrete topology is mutually coupled to every topology on the same set. The possibility for a topology to be coupled with a strictly larger topology leads the author of [258] to the notion of partial order. In [258] more interest is shown in the coupling of topologies than in the situation T1 el U C 7"2 el U for every set U E 7.2 (in our terms - the N-relation). One might think that this preference is conditioned by the following reasoning: the C-relation defines the partial order _< (in our notation < c ) on the family of all topologies on X by virtue of the equivalence T1%C 7"2 ~ (T1CT2 A T1 C T2), and in the subsequent investigations the author considers the cases where 7"1%C 7"2 and (X, 7.2) satisfies the conditions for which it is regular. If instead of the partial order < c we shall consider the relation ~
II
~
~
~
~ ~~ ~
~
'
,~
"r"
~
~
~
~
~3
~
~
r~
:b_,~
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~-.
~
~
'~
-"
~
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,
9
~.
c"'b
2.2. T h e C - R e l a t i o n
79
and so
~-eD(x)
= (~, 2 ) - e D ( x )
c 2-eD(x).
Furthermore, (2) of Corollary 1.3.7 also gives
~-~ n 2 - e ~ ( x )
=
(1,
2)-eD(x) n 2-oD(x)
= (1,
2)-eD(x)
and, therefore, T1 ["] 2 - O ~ ) ( X )
----- 1-OD(X)
= (2, 1 ) - O D ( X ) - (1,2)-OD(X).
The second condition in (4) is equivalent to the first one. Finally, according to Definition 1.3.26, (1) of T h e o r e m 1.3.27 and the first condition in (4),
A c 1-SOZ)(x)=
1-scz)(x)
(there exists a set U c (2, 1)-OD(X) such t h a t U c A c rl cl U) e, > > A ~ (2, 1)-SOD(X) = (1,2)-,5CD(X).
e
Moreover, Definition 1.3.26, (1) of T h e o r e m 1.3.27, the first condition of (4) above and (3) of Corollary 2.2.7 imply t h a t
A c (1,2)-soz)(x)=
( 2 , 1 ) - s c z ) ( x ) ,,, ;
,,' > (there exists a set U c (1, 2)-OD(X) such t h a t U < A < r2 cl U) -,' ;< ;, (there exists a set U c 1-OD(X) such t h a t U c A c r2 cl U = rl cl U) e, ,,
e ;, A < 1-,SOD(X)= 1-SCD(X). Thus, we have
~-soz)(x)
= ~ - s c z ) ( x ) : (~, 2 ) - s o y ( x ) = - (2,1)-soz)(x)
(2,1)-acz)(x) :
= (~, 9 ) - s c z ) ( x ) .
It is also clear t h a t A ~ (1,2)-Soz)(x)=
(2,1)-scz)(x)
~, ,,,
e, ;, (there exists a set U c (1, 2)-O79(X) such t h a t U c A c r2 cl U) < > e, > (there exists a set U E T1 ("12-079(X) such t h a t U C A c r2 cl U) < ?-
e, ,, A < 2-,...qO~)(X)/T1, where the last equivalence follows from Definition 1.3.33. The rest is obvious by Proposition 1.3.34.
D
Note t h a t for the reverse inclusion 2-OD(X) c 1-OD(X), the requirement in (4) of T h e o r e m 2.2.20 for 2-open domains to be 1-open is not superfluous. Example
2 . 2 . 2 1 . Let (X, rl, r2) be the BS from E x a m p l e 2.2.3. T h e n {a, b} c
2-o7)(x), but {a, b}-g 1-OD(X) : (2, 1)-OD(X) = (1, 2)-OD(X) since { a, b} g rl. Corollary
2.2.22.
The following implication holds for a BS (X, rl < c r2):
m
II ~ m
~ -. ~
7-.
& ~ &
II
~
"7"
.,
,
,~, .. ~
7-.
,-~
.,
,~ ,";-" c ~ ~
~
~ ~
7_.
~
m
, ~ ,
m
~
~ ~
~ ~
9
~ II
m
,U-
~
~
~
~
~
~
~ ~
m
~
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~
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tg
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m ~
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~
~
m ~
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IL IL
~
9
~
I-'-4
-.
t~
r-h
r-h ~
;::
.(X, rl, r2) is d-connected. (4) If (X, rl, r2) is 2-normal, then (X, rl, r2) is p-normal and so (X, rl, r2) is 1-normal. (5) r2 A 1 - D ( X ) = r2 A 2 - D ( X ) so that
co (6)
= co
1-Int C1 r2 - 2-Int C1 r2 - (1, 2)-Int C1 r2 - (2, 1)-Int C1 r2
so that 1-C11nt co r2 - 2-C11nt co 72 - (1, 2)-C1 Int co r2 - (2, 1) -C11nt co r2. (7) (X, r l , 72) is p-extremally disconnected e----->, (X, 71~ 7-2) i8 2-extremally disconnected (X, 71, r2) is 1-extremally disconnected e----> (X, r l , r2)
is d-eztremally disconnected. Proof. (1) T h e first p a r t is an i m m e d i a t e c o n s e q u e n c e of (2) of P r o p o s i t i o n 0.1.7 a n d Definition 0.1.14 in c o n j u n c t i o n w i t h Definition 2.3.1. If (X, rl, r2) is not 2-connected, t h e n t h e r e is a set U c r2 N co r2, ~ :/: U ~- X a n d rlNr2 implies t h a t T1 cl U c ~-2 cl U - U, a n d so U c r2 c~ co ~-1. Similarly, one can prove t h a t X \ U c r2 ca COrl. H e n c e U c rl N c o r l so t h a t (X, r l , r2) is n o t 1-connected. (2) A s s u m e t h a t Y c r2 a n d A c Y is any subset. T h e n by (3) of T h e o r e m 2.3.7, r2 int A c rl int r2 cl A so t h a t r 2' int A - r2 int A c rl int 72 cl A cq Y c !
T 1'
int(r2 cl A Cq Y) -
T 1'
int r~ cl A
!
a n d so T1 N % . O n t h e o t h e r h a n d , let Y c 2 - D ( X ) a n d A c Y be a n y set. Clearly, t h e r e exists a 2-open set U such t h a t U cq Y - r~ int A. B y (3) of T h e o r e m 2.3.7, U c r l i n t r2 cl U a n d hence %' int A C rl int r2 cl U N Y c T1' int(r2 cl U cq Y) - r~ int (r2 cl(U cq Y) N Y) -
-
T 1' i n t
(r2 cl rs int A Cl Y) c r~ int r~ cl A
since Y E 2 - D ( X ) , a n d so r ~ N r ~ .
86
II. Different Relations Between Two Topologies ...
(3) The first part follows directly from (3) of Corollary 2.2.8, the first part of (1) above, and (3) of Corollary 2.3.12, taking into account Corollary 2.3.10. The second part is an immediate consequence of the second part of (1) and the implications following Definition 0.1.18. (4) By (4) of Corollary 2.2.8 and Corollary 2.3.10, it suffices to show only the correctness of the first implication. Let F c co7-1, U c 7-2 and F C U. Since F c co7-1 C co7-2 and (X, 7-1,7-2) is 2-normal, there is a set V c 7-2 such that F c V c 7-2 cl V c U. Hence, by (3) of Corollary 2.3.12, F C V C 7-1 cl V C U and it remains to use (4) of Proposition 0.1.7. Assertions of (5) follow immediately fl'om (3) of Corollary 2.3.12. (6) If U E 7-2, then this proof is identical to that of (7) of Corollary 2.2.8, taking into account (3) of Corollary 2.3.12. (7) The proof of the first equivalence is similar to the proof of (8) of Corollary 2.2.8, taking into account (6) above and (3) of Corollary 2.3.12. The rest follows from (8) of Corollary 2.2.8 in conjunction with Corollary 2.3.10. K] Thus, by (1) of Corollary 2.3.13, the conjunction (7-1N7-2 A 7-2N7-1) implies the equivalence (X, 7-1,72) is 1-connected ,+-->, (X, 7-1,7-2) is 2-connected.
D e f i n i t i o n 2.3.14. A topology 7-1 is R0-near a topology ~-2 on a set X (briefly, near 7-2 and 17-1int 7-2 clA \ 7-2 int A I < Ro for every set A c X.
TIN(Ro)T2) if 7-1 is
It is obvious that if IXI < Ro, then 7-1N7-2 ~ 7-1N(Ro)7-2. However, ~iN~j, but ~i is not Ro-near ~j in the natural BS (R, 021, ~22). Clearly, by (2) of Corollary 2.3.13, the N(R0)-relation as well as the (rl,r2) has the (1,2)-(2M,A)-insertion property
(rl,r2) has the (2,1)-(2td,A)-insertion property ==> (rl,r2) has the 1-(M,A)-insertion property. P r o p o s i t i o n 2.4.10. Let A , M and 13 be any families of subsets of a BS (X, 7-1,7-2) and ;g c M . Then the following three conditions are equivalent: (1) (T1,7.2) has the
(i,j)-A-insertion properties.
(2) (7-1,7-2) has the (i,j)-(2X,A)-insertion properties. (3) (7-1,7-2) has the (i,j)-(M,A)-insertion properties.
Moreover, each of the conditions (1)-(3) implies (4) (7-1,7-2) has the (i, j ) - ( ~ , j4)-insGrtioft properties.
Proof. It is obvious that (1) = ~ (2) ==~ (3). On the other hand since ~ c M and 2~ c A for every subset A c X so that ~ c U c F, where U c 7-~, F c co7-j, (2) of Definition 2.4.8 implies (2) of Definition 2.4.1 and so (3) = ~ (1). The implication (2) ---> (4) is also obvious, but the inverse implication does not hold. D Hence it is clear that if 2~ c ,4, then (7-1,7-2)on X has the (i,j)-A-insertion properties e---->, (7-1,7-2) on X has the (i, j)-(A, A)-insertion properties. E x a m p l e 2.4.11. Let X = { a, b, c, d, e, f }, 7- = { Z , { a , b , c } , { a , b , c , d } , X } , A = {{a, b, c, d, e}, X}, and M = {{a}, {a, b}}. Then 2~gA/I, 7- on X has the ( M , A)-insertion property and does not have the A-insertion property so t h a t 7does not have the (2x, A)-insertion property since for the set {f} together with int{f} = ~ and cl{f} = { e , f } , there does not exist a set G E A such t h a t G C {e, f}. C o r o l l a r y 2.4.12.
The following conditions are satisfied for a BS (X, 7-1,7-2):
(1) If M and A are any families of subsets of X and ;g c M , then (7-1,7-2) has the (i,j)-A-insertion properties ,e-> (7-1,7-2) has the (i,j)-coA-insertion properties e----->, (7-1,7-2) has the (i, j ) - ( M , A)-insertion properties.
96
II. D i f f e r e n t R e l a t i o n s
Between Two Topologies ...
(2) If A1,A2 and M are any families of subsets of X, where .41 c A2 and (7-1,72) has the (i,j)-(A/t, A1)-insertion properties, then (7-1,7-2) has the (i, j)-(Ad, A2 )-insertion properties. (3) If M I,A/I2 and A are any families of subsets of X, where .All C .Ad2 and (T1,T2) has the (i,j)-(A/12, A)-insertion properties, then (T1,T2) has the (i, j ) - ( M 1, A)-insertion properties.
Pro@ The proof follows from Definition 2.4.8, Propositions 2.4.3 and 2.4.10.
D
C H A P T E R III
Dimension of B itopological Spaces The notion of a zero-dimensional BS was introduced by I. L. Reilly [216] on the basis of the idea of bitopological disconnectedness as examined by J. Swart [248]. A systematic study of bitopological dimension functions was undertaken independent of one another by M. Jelid [144], [145]; D. (~irid [66], and us [84], [86], [87], [101], [102]. As distinct from [66], [144], [145], the ideas set forth in [84], [86], [87], [101], [102] were essentially based on the notion of bitopological boundaries. The nine functions corresponding to the small inductive dimension [107], [179], [254], the large inductive dimension [45], [82], and the covering dimension [168], [192], [11], [111] of a TS are defined for each integer n > - 1 . Each definition is followed by stating and proving the respective properties of these functions. By analogy with [11] the p-small and p-large inductive dimensions are formulated in terms of both bitopological partitions and neighborhoods in a manner such that for n - 0 p-small inductive dimension leads to the notion of Reilly. In the aboveindicated succession of dimension functions, monotonicity with respect to arbitrary BsS's is proved for the first three functions, while monotonicity with respect to the p-closed subsets is established for the remaining six functions. Further, interrelations of the p-inductive dimensions and their topological versions are considered when topologies are comparable by inclusion or are coupled, 1. If Xl, X2 ~ A, Xl ~ X2, then p- i n d X - 0 implies t h a t , there is a set U c 7"2 Nco7.1 such t h a t Xl ~ U c X \ {x2} since X \ {x2} c 7"1 n CO7"2. Therefore !
I
A O U e (~-2 N co 7"1) \ { •}
i
l
and A \ U e (7-1 n CO "/-2)\ { e }
for the BsS (A, 7"~, 7"~) so t h a t A is not p-connected.
Q
R e m a r k 3 . 1 . 1 3 . If for a given fixed point x E X and any pair (x, A), where A c co7"i, x-~A, there exists a partition T such t h a t ( i , j ) - i n d T < n - 1, t h e n we write (i, j)- indx X < n. The meaning of (i, j)- indx X - n or (i, j)- indx X - oc is clear and, we have ( i , j ) - i n d X - s u p { ( i , j ) - i n d x X " x c X } . E x a m p l e 3 . 1 . 1 4 . Let (X, 7"1) be a TS and B c X be a fixed proper subset of X. It is clear t h a t if 7"2 is the B - t o p o l o g y on X (i.e, 7"2 - 7"(B) - {X} U {A c X " n E 2 B} [14, p. 63]) and if (X, 7"1,72) is l-T1, then (2, 1)-indx X - 0 for each point x c B. L e m m a 3 . 1 . 1 5 . Let x be a fixed point of a subset Y c X , where (X, 7"1,7-2) is a hereditarily p-normal BS. Then (i,j)-indx Y < n if and only if for every i-neighborhood U(x), there exists an i-open neighborhood V(x) such that
V(x) c U(x) and (i, j)-ind (Y N (j, i ) - F r V(x)) < n - 1.
=~ ~
~-'~
"~
IA
~
~
9
m"
~':..~. c-e
~
~!
I
~-~
ce
o-.
=
=- ~
~
~
~
-"
~
~
Z~"
|
~
?,.
c-e
~
~-"
x
i-,,
~-
i,~
~"
~
~-~~
r
~.~
~
~
II
~
9
CT
~
~-~
o
..
o
o
n
II
~
"--"
~
-
(T1,T2) has on X the (i, j)-(2 x , Ti N COTj)-insertion properties). Then (i, j)- Ind X - 0. 3.2.38. The following conditions are satisfied for a BS (X,7-1,7-2)" (1) If 7-1 0
"---"
>
148
IV. Baire-Like
Properties
of Bitopological
Spaces
Proof.
(1) L e t {~n}n~176 b e & sequence of pseudobases for r2 verifying the (2,1)-pseudocomplete property of Definition 4.2.12. Since rlSre, by Definition 2.1.2, r2 \ {~} is a pseudobase for T 1 a n d , consequently, {Bn}~~176is a sequence of pseudobases for rl. If B~ c B~ is any set, then by (1) of Definition 2.1.1, T1 intBn # 2~ for each n = 1, oc. Since (X, rl,re) is 1-quasi regular and Bn+l is a pseudobase for rl, by (3) of Proposition 0.1.15 for i = j = 1, there is a set Bn+l C Bn+l such that 7"1 c l g n + l C T1 int Bn. Hence T1 clBn+l C 72 int Bn for oo
each n -
1, oc which implies that
n B~ # z because T1 C T2 and (X, rl, re) n--1
is (2,1)-pseudocomplete. It is obvious that by rl c r2 we also have re clB~+l c r l i n t B~ for each n = 1, oc, and (X, rl, re), being 1-quasi regular, means that oo
(X, TI,~'e) is (1,2)-quasi regular. Since n Bn 7~ ;g, once more applying Definin=l
tion 4.2.12 implies that (X, rl, re) is (1,2)-pseudocomplete. (2) Let {Bn}~~176be a sequence of pseudobases for r2 for which (X, rl,re)is (2,1)-pseudocomplete. Since 7 c re and each member of re \ {~} contains a member of 7 \ {~}, we have 7Sr2. From this fact we find that {B~}~~176 1 is a sequence of pseudobases for 7 and thus "7 int B~ 7~ 2~, where B~ E B~ for each n = 1, oc. Since (X, r l , 7 ) is (2,1)-quasi regular and B~+I is a pseudobase for 7, by (3) of Proposition 0.1.15, there is a set B~+I e Bn+l such that T1 clB~+l C 7 int B~. Moreover, 7 c re implies that rl cl Bn+l C T2 int Bn for each n oo
1, oc, and (X, rl, re) is (2,1)-pseudocomplete gives n Bn r 2~. Thus (X, rl, 7) is re=-1 oo
(2,1)-quasi regular and n B~ # ~ whenever B~ c B~ and T l c l B ~ + I
c 7intB~
n--1
for each n = 1, oc so that (X, r l , 7) is (2,1)-pseudocomplete.
[3
T h e o r e m 4.2.14. A (2, 1)-pseudocomplete BS (X, 7-1 < ~-2) is an A-(2, 1)-BrS.
Proof. Let {Un}n~ be a sequence of 1-open 2-dense subsets of X and let {Bn}n~176 be a sequence of pseudobases for ~-2, for which (X, T1, ~-2) is (2,1)-pseudocomplete. oo
It suffices for this to prove that
n u~ c 2-l?(x). Let U c ~-2 \ {~} be any set. n---1
Since U1 c T1Ne-T)(X)C ~-2Ne-T)(X), we have UNU1 c T2\{2~}. But (X, 7-1,7-2)is (2,1)-pseudocomplete so that it is (2,1)-quasi regular and thus, there is a set B1 c B1 such that T1 clB1 C UNU1. Furthermore since 72 intB1 r ~ and U2 E 2-7?(X), we have ~-2 int B1 n 0"2 # 2~. By analogy with the above, there exists a set B2 E B2 such that T1 C1 g 2 C T2 int B1 N U2 C U N U2. Thus in a similar manner for any n > 3, we choose Bn C B~ such that T1 el Bn C 7"2 int B n _ 1 n V n C V n V n and, oo
oo
consequently, n Bn # 2~ since (X, T1,72)is (2,1)-pseudocomplete. But n Bn c n=l oo
n ( u N Un) and U E 7-2 \ {2~} is arbitrary implies that n=l
n=l (9o
n u~ c 2-:D(X).
D
n=l
Finally, we give a new characterization of almost (i, j)-Baire spaces different from that given in Theorem 4.1.4.
4.2. Almost-(2, 1)-Baire Spaces
149
D e f i n i t i o n 4 . 2 . 1 5 . A Nj-sifter on a BS (X, T1,7-2) is a binary relation Kj on the family A o ( X ) - {A - U N V ~ 2~" U c T1, V E T2}, satisfying the following conditions"
(1) (2) (3) (4)
A1 F j A2 ~ A1 C A2. For each A c A0(X), there is U c ~-j \ {~} such that U Kj A. (A~ C A1 Kj A2 C A~)---5, A~ Ej A~. For each sequence { A s } ~ - i c A o ( X ) such that As+l Kj AN for every (3O
n-l,
oc, wehave n A s r n=l
It is evident that every Nj-sifter on A o ( X ) is a j-sifter on the family of all non-empty j-open sets [611 and the result of Choquet [611 together with (5) of Theorem 4.1.6 give: there is a n2-sifter on (X, 71 < 72) ---5, there is a 2-sifter on (X, rl < 72) (X, 71 < 72) is a 2-BrS ==> (X, 71 < r2) is an A-(2, 1)-BrS. In the general case, we have T h e o r e m 4.2.16. If there exists a n j - s i f t e r on a BS (X, 71,72), then (X, 71,7-2) is an A-(j, i)- BrS.
Proof. By (2) of Theorem 4.1.4, it sumces to prove that if AN r r~Nj-2)(X) for each oo
n - 1, ~ , then n As c j-2)(X). Let U c r o \ {~} be any set and let us prove that s=l oo
U N( n AN) -~ 2~. Clearly for U1 - U, we have 2~ r U1 n A 1 E ~[0(X). By ( 2 ) o f n=l
Definition 4.2.15, there is a set [72 c rj \ {~} such that U2 K o U1 N A1. Therefore, by means of the same condition and the fact that As c j-2)(X) for each n - 1, oo, one can define a sequence of j-open non-empty sets U1, U2,... such that Ux - U and Us+l r-o Us N As for each n - 1, o0. Thus Us+l c Us+l K o Us N As c Us and by (3) of Definition 4.2.15, Us+l Kj Us. Therefore (4) of Definition 4.2.15 (3O
gives that
n u s r 2~. On the other hand, we have U~+I K o Us n As and by (1) n=l
of Definition 4.2.15, U2 C A1, U3 c A2, . . . . Hence (XD
(:X)
n=2
n=l
(XD
n=l
(X)
O0
n=2
n=l
Theorem 4.2.16 together with (4) of Theorem 4.1.6 implies the more general result than one of Choquet. Namely take place the following C o r o l l a r y 4.2.17. For a BS (X, 7-1 ~ 7-2) , the following implications hold:
there exists a N1-sifter on (X, T1 < T2)
(X~T 1 < 7-2) is a n A - ( 1 , 2 ) - B r S
(X, l < there exists a 1-sifter on (X, q < ~-2)
a (1,2)-BrS
( X , 71 ~ 72) i8 a I - B r S .
150
IV. Baire-Like Properties of Bitopological Spaces
Therefore one can conclude that for BS's of the type (X, T 1 < 7-2) which have a VIi-sifter and a A2-sifter, respectively, all results, obtained for (1,2)-BrS's and A-(2, 1)- BrS's are valid. 4.3. Strong Baire-Like Properties
D e f i n i t i o n 4.3.1. A BS (X, 7-1,7-2) is an (i,j)-BrS in the strong sense (also called (i,j)-totally nonmeager, briefly, S - ( i , j ) - B r S ) if every nonempty /-closed subset of X is of (i, j)-second category. T h e o r e m 4.3.2. The conditions below are satisfied for a BS (X, 7.1 < 7.2):
(1) (X, 7.1,7.2) is an S-(1,2)-BrS ~ (X, 7.1,7.2) is an S-1-BrS. (2) (X, 71,72) is an S-2-BrS :-(X, 7.1,7.2) is an S-(2, 1)-BrS. For a BS (X, 7.1 . (X, 7.1,7.2) is an S-(2, 1)-BrS (X, T1,T2) is an S-(1,2)-BrS ~
(X, Tl,72) is an S-1-BrS.
For a BS (X, 7.1 (1,2)-SBrS ~
A-(2,1)- BrS
-->
159
(1,2)-BrS ~ 1- BrS
r
2-WBrS (2,1)-WBrS
For a BS (X, rl < s r2), we have:
(4) (X, rl,7-2) is a (l12)-SBrS ,e---> (X, 7-1,rD.) is a 2-BrS (X, 7-117-2) is a (2,1)-BrS (XIT11T2) i8 an A-(2,1)-BrS ~ (XITI,T2) i8 a (1,2)-BrS ,e---->,(X, r l , r 2 ) i s a n A - ( 1 , 2 ) - B r S ( X I T l l T 2 ) i s a 2-WBrS ~' > (X, 7-117-2) is a 1-BrS (X, 7-1,7-2) is a (2, 1)-WBrS. Pro@ (1) By virtue of (1) of Theorem 4.4.10, it suffices to prove that if (X, 7-1,7-2) is a (1, 2)- BrS, then (X, 7-1,7-2) is a 2-WBrS and1 hence, (X, 7-1,7-2) is a (2, 1)-WBrS. Let (X, rl,7-2) be a (1,2)-BrS and U c T1 \ {~} b e any set. Then by Definition 4.1.1, g is of (1,2)-CatDII and by Corollary 1.5.14, U E (1,2)-Catgll(X). Hence, by (7) of Theorem 1.1.24, U c 2-CatgII(X), that is, U is of 2-CatgII and by Definition 4.4.18, (X, 7-1,7-2) is a 2-WBrS. Furthermore, if U c 7-1 \ {~;~} and U C 2-Catgli(X) or, equivalently, U is of 2-Catg II,
then by (7) of Theorem 1.1.24, U E (2, 1)-Catgii(X). It remains to use Definition 4.4.7. (2) follows via (2)of Theorem 4.4.10 and (1). Assertions of (3) are immediate consequences of (3) of Theorem 4.4.10 and (2). (4) Let g c 7-1 \ {~} be any set. Then, by (1) of Theorem 2.1.10, U c (1, 2 ) - ~ a t g i i ( X ) ~
U E 2-~atgii(X )
and so ( X l 7-11 7-2) is & (11 2)- B r S ,g-----5, ( X , 7-117-2) is a 2 - W B r S .
Thus it remains to use (4) of Theorem 4.4.10.
[3
R e m a r k 4.4.22. The BS's (R, co < s s) and (R, co < s r ) from Example 2.1.4 are 2-BrS's, and by (1) of Theorem 4.4.21, they also are 2-WBrS's, nevertheless aa -r s and co r r. T h e o r e m 4.4.23. The union of any family of 2-open 2-WBrsS's of a BS (X, 7-1 < 7-2) is a 2-WBrS. [,J us sES and A E 2-CatgI(U ) be any set. Let us show that U \ A E 1-D(U) so that w~intA - 2~ in (U,r~,r~). Contrary: let w~intA -r ~. Since A c U, there exists a set Us c tl such that r~ int A cq Us r ~. It is obvious that A rq Us c A implies A rq Us c 2-Catgi (U), and since Us E r~, the set A N Us E 2-datgi (Us). But (Us, w~, w~) is a 2-WBrS and thus Us\(Ac~Us) c 1-D(Us) so that w~ int(AnUs) = 2~. Since 2~ r r{ int A N Us c r~ int(A N Us), we come to the contradiction. D1 Proof.
Let 11 = {Us}sEs be a family of 2-open 2-WBrsS's of X, U =
C o r o l l a r y 4.4.24. The following conditions are satisfied for a BS (X, Vl < 7-2): (1) ( X I Tll 7-2) i8 a 2 - W B r S
if altd only if each point x c X ha8 a 2-open neighborhood which is a 2-WBrS.
160
IV. Baire-Like Properties of Bitopological Spaces
(2) If (X, q , r2) has a 2-pseudo-open covering t~ = {Us },es each of whose members is a 2-WBrS, then (X, rl, r2) is a 2-WBrS. Proof. This corollary is proved trivially.
D
D e f i n i t i o n 4.4.25. A BS (X, T1 < 7-2) is a 1-strict Baire space (briefly, 1-SBrS) if every nonempty 2-open subset of X is of 1-second category in X. Theorem
4.4.26.
The following conditions
are equivalent for a BS
( X , 7-1 ~ 7-2):
(1) (X, rl, r2) is a 1-SBrS. (2) If {Un}~c~__1 is any monotone decreasing sequence of subsets of X where CX9
Un E 7"1 N 1-D(X) for each n -
1, oc, then ~ Un E 2-D(X). n=l
(3) A E 1-Catgi (X) ~ X \ A E 2-D(X). (4) If {Fn}n~ is any countable family of subsets of X where Fn E
COT1 A
oo
1-13d(X) for each n -
1, c~, then U Fn e 2-13d(X). n=l
Proof. We can omit the proof like in the case of Theorem 4.4.19.
[Z]
C o r o l l a r y 4.4.27. The following conditions are satisfied for a BS (X, rl < re): (1) If (X, Wl, 72) is a 1-SBrS and Y E 72, then (IT, 7~, w~) is also 1-SBrS. (2) If (Y, v-{, r~) is a 1-SBrS and Y E 2-D(X), then (X, 7-1,7-2) i8 also a I-SBrS. (3) I f A c ( 1 , 2 ) - S D ( X ) a n d ( A , r ~ , r ~ ) i s a 1-SBrS, then(r1 intr2 clA, r~',r~') and (r2 int ~-2cl A, r;", 7~") are also 1-SBrS 's. Proof. (1) If Y C X, Y E ~-2 and A c 1-Catgi(Y) is any set, then A c 1-Catgi(X) and by (3) of Theorem 4.4.26, X \ A c 2-D(X). Furthermore,
cl y -
cl ( y
( x \ A)) -
d ( Y \ A)
so that Y = Y cqr2clY = Y n T2 cl(Y \ A) = r2 cl(Y \ A) and thus Y \ A c 2-D(Y). It remains to use once more (3) of Theorem 4.4.26. (2) Suppose that Y c X, Y e 2-D(X), (Y,r~,r~) is a 1-SBrS and A E 1-Catg I (X). Let us prove that X \ A r 2-D(X). Contrary: let T2 int A -r ~. Then r2 int A N Y -r ~ and r2 int A c~ Y c A implies that r2 int A A Y c 1- datg I (X). Since Y E 2-D(X) c 1-D(X), we have ~-2int A A Y E 1-Catg I (Y). But r~ int(r2 int A ~ Y) -/2~ and so r~ cl (Y \ (r2 int A A Y)) r Y, which is impossible. Assertion (3)follows directly from (1)and (2).
[:]
T h e o r e m 4.4.28. The following implications hold for a BS (X, rl < r2): (1)
2-WBrS
4==
(2,1)-WBrS ~
2- BrS
4== (1,2)- SBrS ==~ (1,2)- BrS ==~
A-(2,1)- BrS ~
We have for a BS (X, T 1 % C 7-2):
1-SBrS
~
1- BrS
~
2-WBrS (2,1)-WBrS
4.4. Some Modifications of Baire-Like Properties
(2)
2-WBrS
r
(2,1)-WBrS ~
2- BrS
~
(1,2)- SBrS ==~ (1,2)- BrS ==~
A-(2,1)-BrS r
1-SBrS
~
1-BrS
r
161
2-WBrS (2,1)-WBrS
We have for a BS (X, T1 , (X, TI,r2) is a 2-BrS z---->, (X, T1,T2) is a (2, 1)-BrS z--->, (X, T1,72) i8 an A-(2, 1)-BrS z---->, (X, T1, "/-2) i8 a (1,2)-BrS z---->, (X, T1,T2)is a n A - ( 1 , 2 ) - B r S ,z--5, (X, T1,T2)is a 2-WBrS ~( :, (X, 71,72) is a 1-BrS ,z---->,(X, T1,7-2) is a (2, 1)-WBrS ~ (X, 7-1,~-2) is a 1-SBrS. Proof. (1) Following (1) of Theorem 4.4.21 , it suffices to show that (X, 71,72) is a 1- BrS
(X, T1,T2) is
a
(1,2)-SBrS
--~
(X, 7-1,7-2) is a 1-SBrS ( X , T1,7-2) is aIl A - ( 2 , 1)-
BrS.
Indeed, if U c T2 \ {~}, then by Definition 4.4.1, U c (1,2)-Catgi~(X) and (7) of Theorem 1.1.24 gives U c 1- Catgli ( X ) so that the horizontal implication holds. If U c T1 \ {;~}, then U c ~-2 \ {~} and by Definition 4.4.25, U c 1 - C a t g ~ ( X ) so that the right-hand upper implication holds too. Finally, if U c 7-2 \ {~}, then by Definition 4.4.25, U c 1-Cat9ii ( X ) , and applying once more (7) of Theorem 1.1.24, we obtain U c (2,1)- Catgli ( X ) . Thus, by Definition 4.1.5 the right-hand lower implication is also correct. (2) Following (2) of Theorem 4.4.21, it suffices to show that (X, T1,7-2) is an A-(2, 1)-BrS implies that (X, 7"1,7"2) is a 1- SBrS. If U c 7-2 \ {2~} is any set, then U E (~,l)-Catgii(X). H e I I c e , b y ( 3 ) o f T h e o r e m 9,.2.20, U E 1-~a~gii(X). It remains to use Definition 4.4.25. Assertions of (3) follow directly from (2) above in conjunction with (3) of Theorem 4.4.21. (4) Let U c ~-2 \ {2~} be any set. Then, by (1) of Theorem 2.1.10, U 6 1-Catgli (X) ~
U 6 2-Catg~i (X)
so that (X, T1,T2) is a 1-SBrS (X, TI,r2) is a 2-BrS and it remains to use (4) of Theorem 4.4.21.
[]
T h e o r e m 4.4.29. The following condition is satisfied for a BS (X, T1 < T2): (1) The union of any family of 1-open 1-SBrsS's is a 1-SBrS. For a BS (X, 7-1 < C 7-2), We have:
(2) The union of any family of 2-open I-SBrsS's is a I-SBrS.
162
IV. Babe-Like Properties of Bitopological Spaces
Proof. (1) Let tl = {Us }sos be a family of 1-open 1-SBrsS's of X, U =
U Us sES
and A c 1-Catgi(U ). We shall show that U \ A c 2-D(U) so that ~-~int A = ~ in (U, 7-~,T~). Assume the opposite: 7-~int A -r ~. Clearly, there exists a set Us c tl such that ~-~int A c~ Us # z and A • Us E 1-CatgI(U ). Since Us c ~-~, the set A f-I Us c 1-Catgi(Us ) in (Us,T~,T~). By (3)of Theorem 4.4.26, ~-~ cl (Us \ (A ~ Us)) - Us so that ~-~ int(A c~ Us) - 2~. But 2~ # ~-~int A A Us c 7-~ int(A c~ Us), which is impossible. (2) Let 11 = {Us}sos be a family of 2-open 1-SBrsS's of X and U = U Us. s6S
Since Us c 7-2 for each s E S, by (2) of Corollary 2.2.8, ~-~ < c ~-~ for each s E S. Hence, following (2) of Theorem 4.4.28, (Us, T[, ~-~) is a 1-SBrS ~
(Us,T~,T~) is an A-(2,1)-BrS
for each s E S and Theorem 4.2.7 gives that (U, ~-~,~-~) is an A-(2, 1)-BrS. Since U c ~-2 \ {2~} and once more applying (2) of Corollary 2.2.8, we find that ~-[ < c 7-~ and by (2) of Theorem 4.4.28, (U, ~-~,~-~) is also a 1- SBrS. E] C o r o l l a r y 4.4.30. The following conditions are satisfied for a BS (X, T1 < 7"2) : (1) (X, 7-~,7-2) is a 1- SBrS if and only if each point x c X has a 1-neighborhood which is a I-SBrS. (2) If (X, "t-1,7-2) has a (1, 2)-pseudo-open covering t2 = {Us }~cs each of whose members is a I-SBrS, then (X, 7-1, T2) is a I-SBrS. For a BS (X, T1 < c ~-2), we have: (3) (X, T1, ~-2) is a 1- SBrS if and only if each p o i n t x c X has a 2-neighborhood which is a I-SBrS. Proof. (1) and (3) follow from (1) and (2) of Theorem 4.4.29, respectively. Assertion (2) is an immediate consequence of (2) of Corollary 4.4.27 in conjunction with (1) of Theorem 4.4.29. [3
CHAPTER V
Dynamics of Bitopological Relations, Baire-Like Properties and Dimensions In studying various bitopological concepts, it is of interest to know what types of functions preserve these or other properties of BS's. Since in order to be a BS, the set should have two different structures of its subsets, it is reasonable to expect the function to possess certain additional properties so that it could be used for comparing BS's. It is therefore natural that we first investigate different classes of mappings of BS's together with their interrelationship and then use them to study how the properties of BS's are preserved both to an image and an inverse image [85], [95I. The studied different families of subsets of BS's (X, T1, ~-2) and (Y, 71,72) are of crucial importance in defining different classes of functions of X to Y, while the inclusion and the relations S, < s , C, < c , N, and :
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192
V. Dynamics of Bitopological Relations and ...
k >_ 1 such that ] f - l ( y ) ] , (X, 7-1,7-2) is R-p-T1 ((Y, 71,72) is R-p-T1), (X, T1, "/-2) is 1-second c o u n t a b l e ((Y, 71, "/2) is i - s e c o n d c o u n t a b l e ) ,z--->, (X, rl, r2) is d-second c o u n t a b l e ((Y,'/1,'/2) is d-second c o u n t a b l e ) , a n d co7-2 = p-Cl(X). K] Corollary 5 . 2 . 1 8 . L e t f " ( X , T 1 < N 72) ~ (Y, 71 < N 72) be a d-closed and d-continuous function from a l-T1, 1-second countable, and 2-normal BS X to a l-T1, 1-second countable, and 2-normal BS Y such that for every set A E co~-2 the restriction f A" A ~ f ( A ) is d-closed and d-continuous. If there is an integer k > 1 such that f - l ( y ) l < k f ~ every y C Y, then
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for x C Lj, y E Li. Thus it remains to use (3) ,z---->,(4) in Theorem 6.1.24.
[3
D e f i n i t i o n 6.1.26. A G.lattice s = {L1, A1, V1, 0, ~ , e , L2, A2, V2} is said to be G.complemented if there exists a pair ~ = (pl, p2) such that Pi : Li ---, Lj are maps and x Aj ~ ( x ) = @, x Vj ~ ( x ) = e for each element x c L~. The pair p = (~1, ~2) is called a G.complementation operator. P r o p o s i t i o n 6.1.27. For a distributive G.lattice s = {L1, A1, V1, 0, 4 ,e, L2, A2, V2} the G.complernentation is unique.
6.1. Gosets, Generalized Lattices, ...
221
Proof. Let q p ' - (p~, p~) be another pair, where ~{'L~ -+ Lj are maps such that x Aj p~(x) -- O and x Vj p~(x) - e. Then by GDL2 we obtain ~ ( x ) - e v~ ~ ( x ) - (. Aj ~'~(.)) vo ~ ( . ) = (x vj ~ ( . ) )
Aj ( ~ ( ~ ) vj ~ ( . ) )
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so that ~{(x) _< ~i(x). The case ~i(x)
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6.2. Generalized Ideals and their Variety . . . .
243
i f x 4 Y, then Cli(x) 4 Clj(y) and Inti(x) 4 Inty(y) for each pair (x, y) c Ai x Aj.
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and x = Inti(x) ,z--->, ~i(x) = Cli(~i(x)) ,z----5,~i(x) = Clj(~i(x)) for each x c A i. The proof, based on elementary calculations, is omitted. A more detailed development of the questions connected to bitopological GBA's and quasi measures on GBA's, in our view, is of independent interest also.
6.2. Generalized Ideals and their Variety. Stone Family of P r i m e Generalized Ideals All our further constructions are essentially connected with the notion, introduced by
Definition 6.2.1. A G.ideal (briefly, GI) of a G.lattice/2 = {L1, A1, Vl, 1~, 4 , e, L2, A2, V2} is a pair I = (I1,/2), where Ii C_ Li and which satisfies the following conditions: (1) I f x c I l U / 2 andycIi, thenxV, ycIi. (2) If x E I1 U/2, y E L1 U L 2 and y 4 x, then y c I1 U 12. E x a m p l e 6.2.2. Let A = {A1,A1,V1,q21,(~,~,e, A2, A2, V2,~2} be a GBA and p = (p l, p2) be a G.quasi measure on A. Then it is not ditticult to see that I = ( I I , h ) , where I~ = {x c A , : p~(x) = 0}, is a GI. Moreover, note that the pairs ({a,d}, {c, d}) and ({m, d}, {n, d}) in Diagram 3 are GI's. Since for every GBA A = {A1, A1, V1, qP1, (~, 4 ,e, A2, A2, V2, P2}, the system { A 1 , A 1 , V I , ( ~ , 4 ,e, A2, A2, V2} is a G.lattice, in the sequel we shall consider, in general, GBA's.
P r o p o s i t i o n 6.2.3. Let A = {A1, A1,V1,~91,O,~ ,e, A2, A2, V2,~2} be a GBA. Then a pair I = (/1,/2), where I~ c_ A~, is a GI if and only if {I~, A1, V1, 4 , /2, A2, V2} is a G.sublattice of the G.lattice {A1, A1, V1, 0 , 4 , e, A2, A2, V2} and x 6 I1 U 12, y 6 A~ imply x A~ y E Ii. Proof. First, let I=(I1, h ) be a GI and let us prove that {/1, A 1 , V I , ~ , / 2 , A2, V2} is a G.sublattice of the G.lattice {A1,A1, V1,0, 4 ,e, A2, A2, V2}. Indeed, if x c I1 U/2 and y c Ii, then by (1) of Definition 6.2.1, z Vi y c Ii. It is evident that z Ai y 4 z and by (2) of Definition 6.2.1, x Ai y E I1 U/2 ~ x Ai y 6 I/. Thus {/1, A1, V1, 4 ,I2, A2, V2} is a G.sublattice. Now, if x c I1 U 12 and y c A~, then x Ai y 4 z and once more applying (2) of Definition 6.2.1 gives that z Ai y C I 1 U / 2 ,z-----N,x Ai y 6 I~.
244
VI. Generalized Boolean Algebra and Related Problems
Conversely, let {I1, A1, V1, ~ , I2, A2, V2} be a G.sublattice of the G.lattice { A 1 , A 1 , V 1 , O , 4 ,e, A2, A2, V2} and x E I1 U / 2 , y C Ai imply x Ai y C Ii. Let us prove t h a t the conditions (1) and (2) of Definition 6.2.1 are satisfied. Indeed, if x c I1 U / 2 and y E Ii, t h e n x Vi y c Ii since {I1,A1, V1, 4 ,I2, A2, V2} is a G.sublattice, t h a t is, (1) of Definition 6.2.1 is satisfied. Finally, if x c I1 U / 2 , y E A1 U A2 and y ~ x, t h e n if, for example, we consider the case y c A j , we obtain t h a t y - x A j y and, therefore, y c lj c I1U/2, t h a t is, (2) of Definition 6.2.1 is also satisfied. D It is obvious t h a t if I - (I1, I2) is a GI, then Ii are ideals in the usual sense and the pair I - (A1, A2) is a GI. It is likewise obvious t h a t for a GI I - (I1, I2), we have x E I1 U 12, y C Ii ~ x V i y C Ii. M o r e o v e r , I1 7s A1 ~ 12 ~= A2 and, therefore, a GI I - (I1,I2) is said to be proper if I~ ~ A~. Thus, by (2) of Definition 6.2.1, I - (I1, I2) is proper
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