POLSKA AKADEMIA NAUK, INSTYTUT MATEMATYCZNY
l;t,
DISSET|ATIONTS MATHtrMATICAtr IRO
ZPRAWY MATEMATYCZl\ KOMITET BEDAKC...
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POLSKA AKADEMIA NAUK, INSTYTUT MATEMATYCZNY
l;t,
DISSET|ATIONTS MATHtrMATICAtr IRO
ZPRAWY MATEMATYCZl\ KOMITET BEDAKCYJ
E}
NY
KAROL BORSUK redaktor ANDRZEJ BIAI,YNICKI-BIRULA, BOGDAN BOJARSKI. znrcivrnw crESrELsKr, lnnzv roS. ZBIGNIEW SEMADENI, WANDA SZMIELEW
CXLI R. C. HAWORTH and R. A. MCCOY
Baire
spaees
A I9?? WYDAWNICT WO NAUKOWE WARSZ AW
PANSTWOWE
i,1:;:rr,l.:i :;;.:t: :1
'::
,1
:i
CONTENTS
E-.Besic properties
of Baire
o
spaces
o
Nowhere clensc sets ). D.
If,
18
18
2. Baire category theorem 3. CompLete type properties which imply Baile 4. Minimal spaces
lo lo trl
of Baire spaces
2S
theorems
The clynanics of Baire
l.
Images ancl inverse images of Baire spaces
2. Baire space extensions
F:::?roilucts of Baire
spaees
1. Finite produots ?. Infinite proclucts 3. ft-Baire spaoes 4. Protluet oounterexamPles
36
spaces 38 4L
44
spaoes
3. llyperspaocs and functions
29
.
2. Covering ancl filter aharaoterizations 3, Characterizatiors of Baire spaces involving pseuilo-complete 4. The Banach-l{azul Eame 5. Countabiy-Baire spaces
,. :.1 , ':::
1i}
to Baire $paces 1. Baire spaces in the strong sense.
1. Blumberg type
:
11
l'oncepts relateci
Characteri.zations
;-EF,
8
First and- second oategorY sets Baire spaoes Isolatecl points and Baire spaces
44 4S
Bpaaes
53
56 I}()
60 o+
69 12
ir 'iri
i ;'
I g
i
Introduction
the Baire Category Theorem, versions of which were first
proved in 1897 aniL 1899 by Osgoocl [51] antL Baire l5l, respectively, &at every cornplete metric space is a Baire space (the term Baire I was introd.ucted. by Bourbaki). This theorem is one of the principle thlough which applications of completeness are made in analysis. ({completett space only 1e lr*" ,i*stances one needs to have a m such as the Baire Category Theorem, so that being a Baire lrilr,all that is really necessary. This is tho case in such weli-knorrn as the Closecl Graph Theorem, the Open Mapping Theorem ancl Bound.edness Theorem. llhe Baire Category Theorem is useil to obtain certain topoiogical results'suc]r as establishing that ,*i-dimensional soparable metric space can be embeclde cl in Euclid-
I
space.
Uhe tnown results concerning Baire spaces and relatecl topics aro in variety of papers and books. ft is the purpose of this tract anil organize most of these resu].ts. We have attempterl to inclithe various theorems have occured.. however a number of the t'heorems can be found. in a number of sources, so that we have a reference for every theorem. AIso some of the theorems are ,' have never occured. in a published paper before. ce letters N, Q, and. J will represent the natural numbers, the :sulnbersr and the irrational numbers, respectively. Euclidean eill be d.enoted by 8". If l" is a subset of the space X, then the cLosure of -4 relative to the space Z will be abbreviated- as gLxA, respectively, or just int,4 ancl cll- where there is no
*@
*$
$$
d
g,
I. Basic properties of Baire
spaces
In. this chapter we cliscuss the notions of nowhere d.ense sets. first and. second. category sets and Baire spaces. lVlany of their basic properties are revealed, ancl different characterizations of each are given. Some of the propositions and. their proofs can be found. in l13l or [4]. In conclusion we d.evelop a relationship between isolated" points and. Baire $paces, The results of this chapter will be significant throughout the d.evelopment
of this
J^T Bd.c
8E{f
C{
ffi .:
#,'i
paper.
l': il
l.
Nowhere dense sets
Let A be a subset of a topologicat space Z. Then A is d'en'se i?z X if cIA : X, and A is somewhere d'ense in X rf intcl, + g. Tt .4 is not somewhere d-ense in X then it is callecl nouhere d'ense'in X. PnoposrrroN 1.1. Let N be a snbset, of a space X. Th'en tlue followi'tt'g are
equ,'iual'ent:
(i) rY os nowlt'era d'en'se 'in X. (ii) X - cl-I[ is dense i,n X.
(i;ll) Ior each nonempty open set U i,n X there en'ists a nonempty open' set V i,n X such that V c. U and' V aN :6. Proof. (1) imgilies (ii): I-.,et T& be any open subset of X. Since intcl-ll :8, W intersects X- c1rY. {ii) i,mpl,i,es (iii): By letting V : an(X-cl-l[), we have the clesirecl result. (iil) impli,es (i): If intcl-l[ *A, tlrcn any nonempty open subset of intcl,ltr would. intersect -Iy'. PnoposrrroN 1.2. Let -{ be a fami,tE of nouhere d,ense subsets of X. If LI i,s lacallg fi,ni'te o,t a d,ense set of poi'nts of X, then U.f i,s +towlwre dense 'itr' X. Proof. Let -{: {lf,l aeA)} be a family of nowhere d.ense subsets of X which is locally finite at a clense set of points of X. Let tr be an arbitrary nonempty open subset of X. Since {c1-}["] ae A] must also be
f
-
@ :7 'r
;i,
E{ *l
.1.,
ffi
:..
:::
itu
?o!
'3Si
.$
tr :'i
# &*
i,
:
&
*
I.
Basic properties
y i*ea,tty firite at a d-ense set of points, there exists a nonenpty open set r*ntajaed. in I/ which intersects only f initely many membels of {cl-M" I rt e A}, s*y {ci-}I..}zl : 1 ,2,..., n}. Since each .X-cllY". is a clense open sub'qet n tX-c1]ro)] +6.8y the choice of 8f X, we can see that W :Vo[ i:I -fi2 is open' W c XF, *.2(X-c1-I{") - X- [-l cl-Ar"' Since 1L
*qgt€$
X-cI(Ul|')' PnoposrtroN 1.3. Let Y be a subspace of x, ancllet N be a su,bset of T. in, x. cott'lersely, i,f Y .IJ.:_\r ds ttowhera d,ense,in I, th,en N,is,tzowheye dense i., opu* (or itrense) 'itt, x and N 'is itowltere de+rse 'in x, then N 'is ftou;here
:i:.na
dense
'{clUff;!
f*r.r,t
-*.;j,#*.
,:i+etrf
3tt
t"*it}}e-
t" ;j.
.+-p*{++4 :
tnerefore,
f/
must intersect
'in Y. Proof. suppose that,l[ is nor'here d.ense in Y. Iret u be a nonerirpt}r opel subset of x that intersects Y. Then there exists a nonempty set Tr, is a set 17, *-pen in Y, such t]ILat v c. utY antl 7n-li:o. Now there :6' ojren,in X, such tinat' V : Tf nY. Thus, W c U and' T{n-N lsow suppose that Y is open in x ancl that -M is nowhere d-ense in 5. I,et T/ be a nonernpty open set in Y. Then 7 is open in x. Therefole2 ilrere exists a nonenlpt)- set U, open in -tr, such that TJ c Y ancl [inlY :0. Thus, -M is nowhere clen"qe in Y since t/ is also open in Y' A siuriltlr proof rvould work il the case where I is d'ense in X'
t
Th" next proposition ([3G], p. 154) points out that the finite procluct al subsets is nowhere d.ense if and only if at least one of the subsets is itrelf nowhere d-ense. I{otvever, this is not necessaril;' 111r" for infinite prod-ucts as the following example iliustrates. For each positive integel ti, let x, : [0,1] ancl -l7o : [0,1i2]. rnen fr -N; is nowhere d,ense irr -fi xu 'j:1
.'
while for any fixecl i, il,, PnoposmoN 1.4. 7or eaclt ae A
i--r
in X6' let N" be a su,bsel, of the
is sotnervhere tlense
I "1f3t{i+
5
:t,*1
spctce
Yhrcn, 17 N,'is nomhere dense'in' n X"if u'tt'd' otr'l'11 i'f for som'e Be A, f o€-{
i
itt Xu or cINo + X"lor infitzi'telg mMLy a'e A' Proof. suppose that for each sey', iTo is somewhere Then read. that cl-l[": X" for all but finitely nranX a'-4'
r3f
F:i€
intcL(l/r\'") oe .l
siF i"*f&.f;:s
.€?::t;,
f];Y"
c€-{
[1.Ff*gj ,u",*ri
;qg
&:,1?
tl*
B
is
aeA
wsnhero d,ense {rFFt'-i.
xo'
:
clense
in x"
fJ{iot"tr'") +s' d
2} is a countable regular filter base contained, in 4. Therefore
)
Uo
Let,
+9. X is now a Baire space since ff orc i:2
Urn
(fr lr1. 'i,:1
With only slight moclifications of the above proof we obtain the following theorem. T onpu 2.9. Eaerg guasi,-regul,a,r' coco,nlpa,ct space i,s a Baire spa,ce. The example given after Proposition 1.14 is a metric Baire space that is neither cocompact nor countably sutrcompact. Becalling that the original interest in Baire spaces was sparked. try the Baire category Theorem, it should. be notecl that both the class of locally compact rlausd-orff spaces and the class of completely metrizable spaces include the Baire spaces in the strong sense, the spaces that are semplete in the sense of dech, the quasi-regular almost countably complete spaces, the pseudo-cornplete spaces, the weahly cr-favorable spaces, the quasi-regular cocompact spaces, and the quasi-regular countatrly subcompact spaces. of course, each one of these classes contains the Baire spaces. For additional properties that imply Baire see [?3], p. S3B.
t
&.1
.sF 'ii
3:, t;
*€
P gtr
@
s i,i
Ei*i
.dd
e&
f&
'.: ll:
.*ci
a+ iFr
€3 3*
4. Minimal
spaces
Minimal topological spaces haye been investigated for a variety of properties. rt is our intention to see where the notion of Baire space fits into this area. clearry, (x,{) is a minimal Baire space if ancr onry ttg is the ind.iscrete topology. rlere rve will stud.y the Baire spaces that also have one of the following separation properties: ?r, Ilausdorff , Urysohn, and, regulal llausd.orff. IMe confine ourselves to these four classes because minimai P is equivalent to compact Ilausdorff when P is equal to such properties as completely regular Ifausd.or.ff, normal llatsdorff, paracompact llausd-orff, rnetric, completely normal llausdorff, locally compact lfausd.orff, or zero-d.imensional llausd.orff. tr'or an in depth stucly of minimal topologtcal spaces see [g]. Given a topological property P, a P-space (X,f) is mi,ni,mal P i!., und.er the partiat orcleri-ng of inclusion, g is a minimal element in the se6 of all topologies on .X with property P.
ffi
fs ..:
#r ES
es l',:
.:€, .,;.
i
6,€ai
-
':,1
{Yr
-ff:
g.l
{-a
IL
l:i.,n of nztl U, .!,.
--: i rl.IiLn
!!e
can
.a J/.. ,. ,rieiOr'e
.,:r
i1t"
.
.. iltJ-C€.
i
:Fi1Ce
l-+,'i hr.i-::
ul
;.: i:ilble
;e':
al'g
-ir *tlll1:|It{-'€S.
lrinLlI .:.1:: thg
:'.
to Baire
spaces
2n
is not clifficult to see that a spauce {X,g) is minimalf Lif. and.only if. { ts the finite completnent topology. It (X,9-) is a colntably infinite space with the finite complement topology, then (X,f) is a minimal ?r-space that is of first category. The following proposition gives a criterion lor cletermining when a minimal ?r-space is a Baire space' PnoposnroN 2.10, A m'ininzal, Tr-space (X,,q) 'is q, Baire space i'f ctncl only if X i,s fi'nite or u,ncorttt'table. Proof. Suppose lbat (X,{) is a countably infinite Baire space. Since.r' is the finite complement topolog;', (X,{) wo1lc1 contain no isolated points. This, hon'eYet, is inconsistent with Proposition 1.30. Clearly, ever}i finite space is a Baire space' So, suppose that X is an lncountable set. Let {-4;} be a countable familv of finite subsets of X. N : X-l) Au which is an unco*ntable set anc1, therefo'e, n (X
It
i;i44:t X
-
Concepts related
i133.
l: r
-Au)
deniie in (X,9). Thus, (X,g') is a Baire space. PnoposrrroN 2.11. Let (X,{) be a Ba,ire sgtace and,let.{" be atoytology on, X contained, i.n,q. If there eri'sts a' pe X str'clt th,at {U,{l 7t1U} c{*,
(X,{*1 is a Bu"i,re spctce. Ploof. Suppose lilrut (X,.{*) is not a Baire space. Then theTe exists a nonempty set U ,gu thatis of fir"st category in (X,f*). So t/ : Ll tr, i:r rvlrele each J, is clonecl ancl nonhele d.ense in t.X.{*).It p( T, then each i-o is nowhere clense in (X,.V), so that U wouicl be of first categoly in (X,9-). Tf gte U, then pe -A',, for some ?1' fn this case each Noo(tl -ffr) is no'r,here clense irt (X,{), so that U _ N,: i 1ratu.,(U-If,,)l woulcl i:r be of first categor-v in (X,9). PnoposrtroN 2.\2. Eaer31 m,ilti,mal (7, Ba'ire)-space is fi,ni'te ot' /tL?I,-
then,
cotctr,table.
I-rct (X,,,r') be a countably infinite Baire ?r-space. By Proposition 7.30, (X)9-) contains an isolated, point p. Define
Proof.
-Xii;
-:_tr- :f--{.
7" :.L''r( U{r',1 y, x - {ilil, where l'r, : {A c. Xl p, A and' X -/" is finite} ancl J[o : {t}e{l :J, U ancl pg C), for an)'y€ X-{p}.To see that{" is a topolog)' on X, Iet Ae Nn ancl let (J, Nofor some y, X - {p}. Nox' Av.U e 1Y*' and" lt ze AnIJ, tlren ,4ntr.-V,. Since p is an isolatecl point, {" is properi}' containecl in{. (X,.fo; is a ?r-space "qince each point of X-{tt} is closecl ancl X-{p\e if, for an.v lJe X {p}. (X,{*) is a Baire space by Proposition 2.11. Thus, (X,{) is not a minimal (?t Baire)-space. Tmonuu 2.73. 6,f) is a rttini,m,al (T, Baire)-sgtoce i'J and' on'ly i'f (X,f ) i,s a. Ba'ire m,'itt'imal Tr-space.
i,
J3airc spaces
Proof. Suppose that (Xrf) is a rninimal (?l Baire)-space. Br. prop_ osition 2.12, x is either tinite or uncountable. rf .z* represents ille finite complement topology on x, t'bala (x,f*) is a Baire space by proposition 2.10. Now.V* cf since (X,f) is a ?r-space. Thus, f :f* "since (-{,/) is a minima'r (7, Baire)-space. conversely, if (x,g) is a Baire rninimal ?r-space, it is surely a minimal (?, Baire)-space. There are theorems for rfausd-orff, urysohn, and regular Harisclorff spaces that are analogous to Theorem 2.18. since the proofs of all are similar we will give a proof only for the case of a regular rlausclorff space. This case has an adtled. strength that d-oes not carry over to {Jrysohn spaces or Elausd.orff spaces since rlerrlich [30] has given an example of a minimal rlausdorff space that is of first category, and stepherlson [62] has given an example of a minimal urysohn space that is of first aategory. Let F be an open filter base on a space x. A point e;e x is an adlrere,trt poi'nt of.F tf ne cl? for each Fe g. g conaorges to r if every open set about r contains some member of g. g is urysohn provid.ed" that for eyery grx, if gr is not an aclherent point of g,t]nen there is an open set r/ containing u anil a set Ve7 such that clUnclV :fr. Many characterizations of certain minimal topological spaces have treen discoverecl. Among these is that {x,{) is a minimal rlausdorff (resp. urysohn, r:egular rlausdorff) spaee if ancl only if every open (resp. lJrysohn, regular) filter base with a uniclue ad.herent point is conyergent [68]. This fact inspires the next two propositions that will give us a methocl tor constructing a strici;I5. $'eaker topology on certai:r kincls of spaces. PnoposrrroN 2.14. Let (x,f) ba a Eausd,orff (resp. arysoltn, regu,lar" Eau'sd'orff) Ba'ire spa,ce, and, let F be a norlcow)ergent open (resqt. (Irysohtt,, regular) fdlter base wi,tlt' a unique adherent poi,nt 7t. rf g*: where // : {Uef I ptU} antl "tr : {UvVl TteUe{ and, VeF},"//vJr, th,en {* ,is a Hau'sd'orff (resp. urysolm,, regular H*u,sd,orff) Rai,re topotogy on x tha.t i,s
properly {. Proof. {* is a Baire topology on -x by proposition 2.11. To coinplete the proof refer to l8l ancl lb9l. Pnoposrrrox 2.15. Let (x,t) bo a regular Hau,sd,orff Bai,re space, q'ncl let F be a nonconaergant, open (rospt. urgsohn, regul,ar) Ji,ttey base wi,th a u,n'iqu,e adh,erent poi,nt p. If g :.1/v-{, wh,ere .,V7 : {UY, there exists a dense subset D of- X such that -f In is continuous. It is known [14] that for a metric space X, .X is a Baire space if anci only if X has Blumberg's property with respect to the reals. In fact much more general versions are known (see for example Theorem 3.7). In the next two theorems, we give the proof of a version of Blumberg's Theorem which at the same time incorporates W-hite's result [71] that every semi-metrizable Baire spa,ce contains a dense metrizable subspace. TrrsonnM 3.1. Let Y conta'in an i,n'fi,nite d'i,screte su,bset. Ihen if X has Bl,umbarg's property wi'th respect to T, X i,s a Bai're spaoe. Proof. Let {ys,1J,....} be an infinite d-iscrete subset of Y. If X is not a Baire space then there exists a set U, open in X, which can be represented as the countable union of nowhere dense subsets of X, say U
:
@
from -X into Y as follows: let f (n) :"t/o for each ue X- U, and let' f (n) :'Y^ for each rue U where i,k -- rnin{il *e Naj. Thus, for any dense subset D of X, Jlp is not continuous. LJ
i:1
r
..i'
ir
Ir.
Define a function
/
30
Baire
spaces
Tsnonnu 3.2. Lat X ba a pseud,o-semi,-motri,zable Baire spaee, let T be a second, cou,mtable spa,ce, ancl, tet f : X-->Y be a ftr,netio,tt,. Tlten, thete
a clense rnetyizable su,bsgtace D oJ x such, th,at f lo is cottt'irtuotrc. Proof. ILet d, be a pseudo-seni-rnetric tor (X,g). tr'or each rle .X ancl e ) 0 let U(r, e) : int{ae Xl d,(*, z) < e}. Now {U(x, e)l ne X and. e> 0) is a base forV.It Ac. X ancl neX, then we will say that u is. a heauy poi'n't relat'ive to I if there exists a,n € > 0 such that for every ae IJ(r, e) ancl d ) 0, AotI(2, d) is a nonenrptv set of second categ.ory in -X. L,et Z be tb.e set of all points re X such that for errer.y open set G containing / (n), n is a heavy point relative to /- G). L,et, {G *}ir tre a counten'ists
1
*a.{
"t"
(
able base {or Y, and- let E,,be the set of all points in/-1(G*)whieh are not hear'\' points relative to f-t (G"). To see that -tr, is of first category in X let An be the set of all points rrf-t (G,) such that there exists an e> 0 is of first category in X. By Theorem 7.7, An so tlrat U (r:, e) ^f-'(G,") in X. L,et, Bn be the set of all points in f-r(G)-An is of first category that, are not heavy points relative to f-1(G,). Now .B,, is a nowhere d.ense snbset of X. Ilence, -8,, is of first category in X since 8,, - Anu Bn.
Thus, X-Z is of first category in X since X-Z - j U*. Since X is i':r a Baire space, Z is dense in X. For each !/u Y,let {G"(y)}ff:, be a countable base at g x,ith G**r(U) fol each rz. Also for ever;r se Z and. ne N,let W,(a) : f-t[G"(1q2111. G,,(y) BJ' Zoln's lemma, there exists a maximal pairwise disjoint colrectinn %, of sets of the torm U(2, sr(a)) where bhe iollowing are true: (a,)
z
eZ,
(b) e'(z) < $., (c) if re Ulz, er(e)) and d > 0, tiren tI(n, 6)nT{/r(a) is of category
in X.
:7r,..rf;: (i) Z" c. Z. (ii) 2,,-r = Zn. (iii) e,, is a function trom Zn into the positive reals. (iv) For eaclt ze Zn, e,,(z) 0 such that for eYer)' n ',U (z'' 6a@)\, and d > 0, fl(n,6)nW*{z) is of second- category it X' Let e3*t(ql (z))tr < min{dr(a ), ex@), 1l(2i{+2)}. If Bo*,(a) lU {:, 'n(a)) - clU 8, "*+'
:
i*oonempty,thenbyZorn'slemmathereexistsamaximalpairwisethe where disjoint coliection Un*r@) of sets of the form U(n, e*r(n)) following are tt'ue: (1) re V*(z); q\ U(r, ur+r(r)) c Bapl(a)i (3) e1*r(r) < 7l{zlt 12) ; (4) if s , U{*,r7,1r(r)), and- d > 0, t'hen U(s, category in X;
d)
nW*a@) is of
second
I-ret
'ir',,
&l!i{sl H;.{s} qn'rll.
for all
Zr*r@)
: l*l
and for each o< Z**r(z)
(I(n, ur+'(tr))
'
Qlo-r@)lv{a}
le1.,
V t *r(n) :
fi
Z'
t
in u{nr"r+r(#))' I-'et {*+J4 l),Zn+t@) and 9{o*r@), Zx+r 7 z"Zk nr4k
*'
W r,+r(n)
o
tJ
(n, en+t@)\ n
As before we can see that lro*r(n) is d.ense
:
{Tzn-r(ir)
%t
'
| ueZ1,*r{z)),
{**r:
g lwo*r{a}vl(Io*r(2, +t: seZ4
I
e*-r(z))}]'.
therefore have ind.uctivell' d-efinecl seq'nences {Z,rl ne X}' {t"l neN), {Unl n.-}r}, and {f,,1 n',-47} so that for eacln n'-tY, properties (i) thr;ough (x) iisted above ate true' I'et D : \){Z*l *re t[]' It can tre a Baire space' seen that D is clense in f-jle)ot,"l ??'€^7)' Since X.is *" frroo the.t (l ll)qr; ri,. -l/fis dense il-X, so that -D is dense in '(' 11re
"LetaeD.Thenthereexistsapositir'eintegertr;sucht']]nahz,Zn.
fl
i $i 1,.
i'ui irii'':; 1$l
r
Q,,(nt,
*r)
: ll zeZil,
ei,{n')
-
Pi,@r)l
ii-. i .il
,,.,,+& ;;rsii
ao
Bairo
spa.ces
for each (h, fr)e D xD. rt is easy to see tb,ah gn is a continuous pseud.ometric on -D bouncled by 1. -D is a llausdorff space since Zn< Zrr*, for each n e -lr. r,,et ze D and., be any nonempty closed subset of -D such that zS A. Then there exists a positive integer k such that, U(a, eo@)\nA : @. Thus, tor each ae A, Qr,@, a)
: )
lpfr(a)
neZ7,
rfence, int
pn@,e)
-
pfr(a)l >- lp'n@)
> 0. Now g(r, 4
metric on D.
:
- sf,@)l :
fi,fr
*€';
Xer:
*!g eo@)
>
0.
n*ro,a) is a cornpatibie
Cono:,r,Eny 3.3. tret X be a pseu,d,o-semi,-metrizable sgtace, antl let y ba a second' cauntable space whi,clr, contains an i,nfi,nite d,isu.ate su,bset. rken x 'is a Badre space if and, onlg i,f i.t has Bl,umberg's property wi,ilt, respseot to Y. conor,r,Env 3.4. Eaary pseud,o-sent'i-metri,zable Bai,re spoce conta,ins
a
tq
bsq
*Iq i*e
iis -j
t*t
effi Fsq ilTi
is
r;
d,ense metri,zable subspaae,
Many interesting.spaces have a pseudo-base that is the countable union of disjoint families. The next theorem [?1] gives a Blurnberg type theorem for such spaces. Trrnonnu 3.5. Let x haae a o-d,i,sjoint psaudo-base, and,l,et T be a second, countable space wh'ich conta,ins an i,nfi,nite d,i,suete subsat. rhen x is n, Baire space i,f and, onlg if i,thas Blu,mberg,s property w,ith respectto T.
Proof. I-tet g : ()
,1.
:r
,',
:.]:
,w.$
{*,,1 m
ry *)Fen
Eei
Characterizations
,
Theng:t-l"lfisabounded'lowersemi-continuousfunctiononU vn x, g carr easily which is continuous at no point of u. since u is open on x' be extended- to a bou-nd-ed- lower semi-conJinuous funetion Trnox,Eilr 3.9. The folloni'ng are equ'iaalent for a space X: (i) X as a Bai're sp&ee.
(ii)IfCisafami'lgofl'oaersenai.comt,ilt,uotlsfunct,ionsonXsuclt'that i, x tha ia {f tnli f , c} i,s bound,ed aboao, then for eaarg nonem'pty u of x tltero enists a nonamTtty o?ten subset Y of u and, a pos'i' tiue i,nteger k such that i'f y o} :
LJ
l){C,sl
ne C}l f (n)
f***
>
"1. Let p be a point of continuity of /. Then there exists an open set I/ containing p such that/(Iz) is contained in the open inrerv^t (I@)-i,f@)+t). Therefore, U: (n {Ce6l pr}})n7 is an open subset of X containing p. Now any member of € that d.oes not contain ? cannot intersect U. Therefore, U intersects only the members of V which contain p. Ilence, {4 is 1oca1l5' finite at p.By Theorem 3.8, the set of points of continuity of / is dense in X. (Lii) i,m,pli,es (i): Suppose X is not a Baire space. Then there is an open set U which can be expressed as the countable union of nowhere
of X, say U : ,j ,0,. For each i,,let Ui : C[- U cMfi. j:t i:r Now any open subset ol X contained. in U must intersect each Ui. Therefore,{X,U'Urr...}isacountable point finite open cover of Xwhich is not locally finite at any point of Lj Or. d:r As is often the case for characterizations of Baire spaces, Theorem 3.10 has an analog for spaces of second. category which can tre proved. in a similar manner'. Trnonsu 3.I7. Ihe followi,ng are equ'iaalent for a space X: (i) X os of socond, categorg. (IiJ Eoery poi,nt fi,ni,te open coaer of X is locall,y fi'ni,te somewh'ere. {iii) Eaery countable poi,nt fi,ni,ta open caaer of X i,s locally fi'ni'ta some-
fl
dense subsebs
where.
Many topological properties are d.efined. or characterizecl in terms of a filter characterization of quasi-regular Baire spaces [42].
filters. lYe now giv:
,.j.f
'r:,*!;
ffi w
,.
@
k
',&
w
&
III.
Characterizations of ,Baiie
sDa.eos
space X is li,ghtly com,pa,ct (also callecl feeblg compact ot weakly if every locally finite collection of open sets of X is finite. fs6ki i32l has shown that a space is iightly compact O ** only if for every /.A. Thus, d.ecreasing sequence of nonempty open sets {Ar},
A
*+yed.
f ir
$et
E$e aJ
conq)&ct)
]clUt
we can -qee that every quasi-reguiar lightly compact space ('Y,9-) is pseudocornplete by letting 9t :{ for each z. Tnnonuu 3.12. I.f X 'is a quasi'-regu'la1' spece, then tlrc follomitzg are equi,rale,nt:
tit X is a Baire sqa,cc'
(i1) Et:erg poi1fi fitti,te open fi'ltar basa F on
*i'g :*3ity i{,?F€.i
*g .J,
Sei;r-
!-+t. i+lt]ct ,ffi€€,.
ss*tF t'
b
ar.
sft*?e s.''3
1r"-
:&.*r€-
i's locally
fi'ttite at a d'ense
of lF. (ii|) Eaery countable, poi,nt fi,n'i,to, regular open fiLter buse I on' X 'is tocatly fi,n,ite atr a d'ense set of ltoi'nts of I F. (iy) Euery cotmtable, poi'nt fi,'tt'ite, regu,lar open, fitter base F wltliclu ,is not local,lg fi,nite at any pai'nt of l)F has an ad'herent point. Proof. It follows from Theorem 3.10 tiia,t (i) implies (ii) since [J.F is a Baire space. Clearly, (ii) implies (iii) and (iii) implies (iv). To see that (iv) implies (i) suppose that X is not a Baile space. Then there exists a set O, open in X, which is of first category. Therefore U is nol, lightlv compact. Let {Wr} be a countably infinite, locally finite coilection of nonemlity open subsets of U. Now each W1 is of first category in X. For be a sequence of nowhere clense subsets of X such that eac r il, lut {ffu,r} Wr: \)Wtii also let Wrp:0. lSow, for each z, there is a sequence
set of Ttoi.nts
J:L
{70.1} of nonempty open subsets of X such j, cl I/;.;, i c VLj. For each i, and j, clefine
ahich
*scrn r*v*d.
X
(Jr,j I-,et
7 : {Ut,il i2
1,
that
Vo.,r,
:
Wi and', for
each
:( rj r*,,)-( n:0 a k:O U crw;ai-,,,p). k:7 j > 0\. 7 is
acountable, point finite, regular open l)F and has no ad-
filter base which is not locaily finite at anypoint of
herent, points. , Another characterization of Baire spaces in terms of fiiters is that the set of adherent points of every countable (or point finite) open filter base ri-ith empty intersection is nowhere cLense (see [4?])' r,*g.rs. &el$iaa-
at S*ire
raa"*
3.
of Baire spaces involving pseuilo-complete spaces To see if a space is a Baile space, it is sometilnes convenient to check
Characterizations
oniy certain subspaces. Proposition 1.28 gave the following ProPosit'ion [3].
u.s such s, sub"space as does
aa
i,blg1
Ba,ilo spaccs
Pnoposrtrorq 3,13. In eaery qx{a's'i-regula'r space X thore a're open (possemply) subspaces Xo and Xn such that
(i) X"nXn:6, and, XuvXa'is d'anse in X; (ii) .Xp 'is pseu,do-complete; (iii) and, oaerE pseud,o-compilete su,bspace of Xo'is ttowhere rtrense in Xo. Iu,rtrhermoro, X i,s a Baire space 'i,f and, only tf Xe i.s a Batre s'pa,ce. Proof. Let Xp be the union of all open pseud.o-complete subspnces of X and let Xn - X-clX". Conor,r,mn 3.I4. Let
com,yi,ete sflaces. Then
X
X
be
a qu,asi-regular
cou'n'table tr'mion of psetttloif X i,s a pseud'o-cctntplete
i,s a Bai,re space i,f un'd only
space.
Proof. Suppose t'}l.at' X is a Baire space. Then X2 is a quasi-regular Baire space@which is the countable union of pseudo-complete spaces, say X, : p, Xn.If there is an'i such that intclXt *@, then intclXn is
g
1:
Ii
=
{:: .{J'
ii:;
pseud.o-complete since clXo is pseud"o-complete. This contradicts part (iii) of
Proposition 3.13. Therefore, intclX n: fr for eachn.Ilencc fi tt, - e1,T,,) is an empty d-ense subset of Xo, so that .X, is empty. 'L:7 Tlreorem 1.24 gave necessar), ancl sufficient cond.itions for a dense subspace of a Baire space to b; a Baire space. The next proposition [3], whiclr follows frorn Theorent 7,24, characterizes certain Baire r{paces in terms of pseud.o-compiete extensions. PnoposrnoN 3.15. Let X be a guas'i-regal,ar space sttch tltat any psendocomplote sabspace of X i's mowhere d,ense in X. The followi'ng properties are equ'hailent: (i) X zs a Ba'ire sp&ce. $\ nor eaery pseu,d,o-complete space Y suclt, tlrat X i,s d,ense in, Y, each Gu-subset of Y which, is conta'ined in Y -X'is nowhere d,ense,i,tl, T -X. (iii) 7or some psendo-complete space Y such, th,at X,is dense itt Y, each,Go-snbset of Y wh,i.clt, 'is co'nta'ined'in Y -X i,s nowhere d,ense,in y-.f. Proof. For any pseud.o-complete space Y such t1a.at X is d"ense in Y, the set y-X is d,ense in Y, since X contains no open pseud.o-complete subspaces. The proposition now follows immediatel;' from Theoretn
h:
g
nd
?t
.E
ir
*I,
1t.i1, 0,1
::t
4. The Banach-Mazur
game
Arouncl 1928, the Polish rnathematician, S. llazur inventecl a rnatirematical game nolv known as the Banaeh-Ma"z'tu' ga,nxe [53]. Originalll' this game was to be played" on a closecl interval in the real line. The cLe-
III.
Ej. H+ff. &,a€$
'WE*l-
Srfs il
tr'l'al &g.es,
i
1,S
!i,9
*f
5.
[f0,] #n$e t9l
i'r..!r &&4*"i ::
5@Ei€*.€s
sr'' _a
sT, _d
f.' {s t', E l€te *tr*lx1
**ft,th-
les:lr S:r$e-
Charaotelizatiorrs
of Baire spaces
39
.qcription of the game that will be giYen here is generalized liy playing on any topological space. TLet x be an arbitrary topological space, and. Iet %be a collection of subsets of X such that for each Ue Q/,int'U +=6, c ]2. and. for each open v in x,,there exists an element u e Ql suehthat [/ game B) G(C, x. The is union whose of x T.tet A and. B be disjoint subsets Ut choose sSs (B) alternatelS' is played as follows. Two pla5'ers (l-) ancl
fram,2/ such that
U*r-
Uafor each
i.
Player (-4) wins
if -an{f^) Ut)
* 6i
othetwise PIaYer (B) wins. The immediate question is whether or not one player, by choosing his intervals jucliciously, can insure that he will win no uretter hox- his opponent plays. The answer to this question gives us a nice wa-l of charactertzing spaces of second, category [26] and. Raire spaces t34l' 135] through the use of game theory terminology' A svtaco X i,s of second, category i,f and, onlg i,f player (6) does mot hat:e a winn'ing strategg for the game G(X,6) ooith (X) plnEi'rt'g first' x i,s a Baira sTtuce if and onlg i,f plnyer (6) does not haae a u'itttt'irt'g strategg Jor the game G(X,@) wi,th (6) plal1i'ng fi'rst' \trre night point out tbat' X is o-favorable if and. onl;' if player (X) has a winning strategy for the game G(X, O) rvith (CI) playing first. Thus, being a-favorable obviously implies being a Baire spaee in this setting. To translate and. ploYe the above st'atement we need- to introdnce some notation similar to that used. in [34]' If I is a pseud-o-base for a space
X, Iet g(g):{f: 9-91 f(P) eP for each Pe7} ancl let g"(g) : l/, U 9"-->9i /((P.,..'.,P,,)) c Pn for every €,,"',P,,)eQ" ancl IL:7 for every n|. For PeQ arLil f, ge9(9) let [P, f ,97r: g(P)' X'or i]'I, let [P, f ,g)rbe f (lP,f ,g];-r) if t' is even ancl g(lP, f ,7f*t) if t' is od'cl' For P : (Pt,...,P,,)r3n a,nd. /, ge9*(9) let LP,f ,gltr: g{P)' For 'i)1, tet [F, f ,tJli be /((Pr, -..,P,,,|P,f ,g]:'' "',lP,f , gf-r)) if rl is even and g((Pr, ..., Pn,lP, f , gl\, "', LP, f , il;-)J if r' is ocld' frrp'.sRp11 3.L6. Tlto fol,lotoi,tzg aro egu"iaalent
for a
(i) X zs a Bai're spa,ce' (il) There en,ists aTtseudo-bctse S for x suchtltatfor
space
X:
aatE ue@ (Lf ,9", a fu'nct'iotr' ge 9{4} (]lU,f ,sl; +6, r'esp')' (ge9*(9J,resyt.) sttch that ?rlr,f ,lfr*, (iti) Let I be attpl lsseud'o-base for X' The+t' for *n'g U e I (U e 9n, resp') g'9(9) (oe9+(8)' and, f eg(g) (f ,9*{9}, r'esp.) tlt'ot'e enists a functi'ort' (! lU,f ,{t1; +@, resp')' resyt.) such that.Q tU,f ,ili*O
t'esp,) and f , g(g)
(f , g.Y)
resp'J there enists
Baile
spaces
(iv) There en'ists a pseudo-base Q Jor X su,clr, that aaery point fi,tr,ite of X conXa'ined, 'in I is locnl,l,y fi,tr,i,te at a clemse set of poi,+tts
pseutlo-couer
,in X. (v) Let I of
X
:6
*x
I
: U l-r. lYell-order g ancl define a rnember f of 9(9) as d:T foilows. Tor P e I stitih P nLI : A l,et' f(P) :?. 3or P e Q tvit]n P nU + O let,/(P) be thefirst mernbei of g containecl inP-{, where n, is the least integelsuchthat PnNn *0. Nox A t,,-i-u) :0. Uence, fl lU,f ,glt r':i i-t .f,
w
be any psettdo-baso for X. Tlten, any poi,nt fdni,te pseud,o-coaer i,s locally fi,ttite at et clense set of poi,nts i,tr, X.
contai.tuecl,in
Proof. \\te will prove only the case involving 9{9). All irnplications will be establishecl by the contrapositive. Clearly, (iii) implies (ii) and (v) irnplies (iv). To see that (ii) impli:s (i), suppose tll.at X is not a Baire space ancl let Q be any pseuclo-base for f. Then there exists a membel a of I wirich can be expressecl a-q the countable union of nowhere dense subsets of
,.{,
4ai.
,l
{
sl
sa;' U
for each g e g(9). (r) intplies (iii): Suppose that' (iii) is false. Let
X,
I
eg(g)
I
.,{-
EJ
be a pseudo-base
9(9). Brt- znf-clmi.tt, of ord,er n rve shall rnean a nested seqr.ience Gr - G, = ... = Gr," of 2rr, sets belonging to I sr-Lch that G, c I/ ancl f (Grr-,.) : Grr for each
for
Ue
and.f
such that
{l lLr,f ,lli:A
for
each ge
i:
7, 2, ...,'u'. -Lt/-chain of orcler ii f ft is a continuation of one of ord.e:: 2m terms of both chains are the same. Let Y, be the set of all families of /-chains of ord.er one such that if '!lu T' then the collection of all.qeconcl elements of the/-chains in y is pairwise disjoint. Y, is partially orclerecl by set inclusion. Bv Zorn's lemma Y, contains a maximal element, sa--v firr. Let U, be the union of all the second. elements of members of .FIr. Then [/, is a dense open slLbset of [i. Proceerling by inil"uction, suppose a farnil; Xn of /-chains of ord-er ri has been so clefinecl that the set of all 2rz terms of elements of In is pairsise clisjoint and. their union is a d-ense open subset of U. Iret Y,, , , be the set of all families of /-chains of ord.er iz *1 which are continuations of /-chains in ?,0 such that, tt y e Yn*r, then tlre collection of all 2n+2 elements of the/-chains in gr is pairn'ise disjoint. Y,,-, is partiall;r orderecl b-V set incllsion. Again by Zorn's lemma Y,*, contains a maximal element, sa; ?r,+r. Tnt U,ra1 be the union of all the 2tt,+2 elenents of mernbers of 1r,,1. Then Ur+, i"* a d.ense open subset rr,
if the first
n is containecl in f] Ut Then there is a monotone clecleasing sequence {Gr} of elements of / containing r such that G, c L- ancl j(Gu) : Gz; for eacb 'i. Define an element g at 9(9) as foiiows. Let g(U) : Gt ancl g(G";) - Gzt+r for each al
of ['. To
see
that
{l Ut:6
suppose that
*:
s
q
e,y
fr
*
$l
#
{:
.{j
:s
.r.!
*
+
*trl
III.
f*le #l**s
+"?5.:f
r
**13L3
i {s} reee
Fg9 6st* 1! a.c
*8 ffisE
isl ',,.
lea*e
of Baire
spaces
41
otherwise let g be the iclentity function on 9. Therefore, r is contained€ in O lU , f , gla which contraclicts our original supposition'
"Fo"
each n,,let, Qnbe the collection
of all2l?, terms of eiements ot Pn.
Q, is a pseuclo-cover fol f/ sinee each Q, is a pseud.opn+t is a concover for u. Note tbat un : l) Qr, Since each member of tinuation of some member of nr, Urr+tc U*fot eachn' Thus, Z is point finite. Ilence, any open set that intersects f/, intersects infinitely manJr point of U' Note menrlrers of 4/. Therefor:e, [/ is not locaily finite at any (ii)' (iv) implies that a sirnilar argument will show that (i) imltli,es (v): Suppose that (v) is false, I,et 9 be a pseuclo-base for j and let ,2/ be a point finite pseudo-cover for X containecl in Q which is not locaily finite at an-t point of the nonempty open set J/. Thus, uu{r} is a point finite open coYer of x which is not locally finite at any point of I/. By Theorem 3.70, X is not a Baire space' A theorem analogous to Theorem 3.16 can be stateci tor spaces of seccncl category ra,ther than Baire spaces (see [42])' clearly, ,//
.
Characteriza,tions
: l) n:l
..'
ipi. f;_ '-:11
seh nd,er
&.,ef
w31
e$5 S*jo
F"-
pcr6e
f *i] &se
wet
&,ea ,ti, .
i'*., t*&e
h'**t ),
, Lr;.
iS4 i,'Sn
5. CountablY'Baire
spaces
Theorem 3.10 together with Theorem 3.16 suggest an ir:vestigation of the foliowing notion. A space X is a cou'ntablg-Ba'ire'space i{ there point finite pseudoexists a pseud.o-b ase I for x such that evely countable ofx eontainecl i'. I is localiy finite at a d.ense set of points in X' "oo."B-r* Theorem 3.16, eyer-]. Baire space is a countably-Baire space. conve}.se is not l\/-e rrill now present an example to illustrate that the rationals Q with the usual true. L,,et X : n Q", where Q" is a copy of the o<sl open sets topolog3.. L,et *'have the bio-procluct topology; that is, basic where U" form n;t(U"), arl the countable inter.cections of the sets of the knorvn that the is open in Q" ancl z" d-enotes the ath projection. It is well opeir continuous image of a space of seconcl categor;r is a space of second.i,t"go11- (Theorem 4.1). Since the projection maps, even for an lto-prod'uct ]-'s6 *p*Ju, are continuous ancl open, it follows lhat' X is of first categor5r' intercountabie all taking 0 clenote the pseuclo-base for x clefinecl by leait one of these Uois the sections of eets of the form z;t(U"), where at subfamily of 4' Define interTal (0, 1). \'ow suppose that Q/ is a cou-ntal:le for all the members ind"ices cogrclina,te all the of 6 < **, to be the supr.emum all of Q' Then let P is not factor of ?z such that the projection into that : a{ p' and ever'r' for (1,2) be the basic open set d-efined' by: z"(P)
fi"\P):Q"foran}r0>B.Ilence,Pmustbed-isjointfrornevervmembel
10
Baire
spaces
of q/) so th.at al could not be a pseud.o-cover of x. Thus, g does not contain a countable pseud"o-cover, so that x is a countably-Baire space. Without adclitional properties there is no relationship betrveen the notions of counJably-Baire and seconcl category since the disjoint topoIogical sum of the rationals ancl a space consisting of a single point is of second. category but not a countably-Baire space. The above example also shows that the irnage of a countably-Baire space und.er a continlols open Inap need" not be a countably-Baire space even if this image is a separable metric space. This is because Q is not a countab])--Baire space. Now let Y be the ilisjoint topological sum of Q and x (definecl in the previous para,graph). Then Y is a countabl5.-Baire space for the same reason t'},.at x is - no countable subfamily of gqv7 coulcl be a pseucLocover of Y, where 9q is any pseud.o-base for Q, and g was clefinecl in the previous paragraph. This example shows that an open subspace of a countab15'-3*ir" space neecl not be a countably-Baire space. lYe rvill now investigate when the concepts of Baire spaces ancl countably-Baire spaces are ecluivalent ancl list some of the '(Baire like" properties of countabl5.-Baire spaces. A space is saicl to have tlne cou,ntabla cha,itt conili,tion if eveq' clisjoint family of nonempty open subsets of -{ is countable. Pnoposnrox 3.17. rf eoery pseu,clo-base for x conta,ins a cortn,table
pseud,o-coaer, then
X
ltas th,e countuble ch,a'in canclition. Proof. srippose t'hat' Ql is an uncountabre collection of pairwise clisjoint open su'bsets of X. ILet -:// be the set of all pairwise clisjoint farnilies of open subsets of X which contain oil, and. consid.er -r'l as partial\. orrlerecl b)'set inclusion. By Zornts lernma, //lnas a rnaximal element {LT":a, A}. For each ae a, let %obe the famiiy of all open subsets of f/o. clearlj, \-)q/" is a pseuclo-base for x, ancl every pseud"o-cover of x containecl
F:r
,$.
t*
;
i
:r
.*
3:S
ttri.
i !.
ss
g
l
'i $
j
s:$
*,5
{5
.&i
,:|
t.
deA
t"
9,
Ql"would. be uncountable.
PnoposrrroN 3.18. Eaer1tr pseu,do-base el'isjoint pseudo-coz,er of th,e space.
# for a space cotttcr,i,n.s a prtir,u:ise
Proof. Let' I be a pseud-o-base for the space x. r-rct .,// he the set of all pairwise disjoint subfamilies of 9, and. consider .// as partially st6.r..t by set inclusion. By Zorn's lemma, ./,/ ]nas a naximal element ?/. Suppose Ql is not a pseud.o-cover of x. Then there exists a set 7, open in x, suLch that Vn{l)"U): O. Therefore, there is a set Pe7 w]nich is conra,inecl in T/ and., hence, intersects no eiements of Qt. But this means that elv{p} is a member of '// which is strictly larger lhan 0// a contradiction of tiie rnaximality of 42. The next proposition follorrs imrnecliateh. flom propositions ancl 3.18.
8.1?
w
s
rfu
&
s
5
#
siJ
'*:
gia
q'l 5:
III. 53n&t'F.
rthe lsKr-
its r3rle j.i311X
$ge eire I ilt. iT}}A
#s5he
sett.
:l
.$tt -
rs$-
*int
.,. *&fe s'"€se
#nfs
ffiw. *l s!.
*$g{-
is*d
w*sc
L;;."y and, g: B-+? be continuous 6-o7ten is
i;t
34
Thus, X is of first category since f ,lt, "n"". Baire spaces. tror example, rrowever, d-open functions d.o not preserve Iet
into T is
r,,I
d-open,
furct'ions, and, d,efi'ne E: x x8+y xT bE x(n,il:(f(*),g(s)) for ;a,ctl ne X and, se B. Ihen, F ,is a continuous 6-oTtenfu,nction'. Proof. T-let B c f xT, and suppose that intclJ'-l(B) +g. Then, there exist sets u and. 7, open in -E a,nd B, respectively, such fiiat u xv c intsl-tr'-1(B). Since/ and g are d-open, intcf(Tl) +g, and intclg(Z) *@. Let (y,t), [intcl/(U)] x fintctg(V)J, anil ]et M atd. -U be open in 7 and. ?, respectively, such that (g,t)e M x-tr[. Since yeclf(U) anct teclg(V), Mof{U) +9, and" -lln SVI +fr. Let ueTJ and. se I/ so that E(n, s) : (tt*1, g(s)) , M xN. Since / is continuous, U xV c. cl_F-1(B) c
,{}
#T
s
& ::
xs
#
TI
ffi
i
IV, The clynamics of Baire spaces
:,,:
41
.l t::
s
ftt
&ere
ffqlF raf*
:
ffion-
:. .!.
*Fkn
$*al M.-$.
suX :' :.
l
Xi*A's l:1"
:
'
'tr:t: ;"ir'
ry ffF$' &Ka ir.
W* a$pe &en w@s" jl q*!*g+'
sdftrl :ij'' t''t'
- ii'F,
lsF sgl F$erl
!T{* &st' Et\ srL-
clF-1(clB). Ilence, I{n,s)€clB^(Xt x-l[) so tb'at {y,t)ecIB. Therefore, tintcU(U)lfintctg(7)l - clB. E is d-open since int cIB +6. Cleatly, I is continuous. Pnopgsrrrorv 4.9. Not etserg metr,ic Baire space 'is the cont'intr'otts \-ogtetr, i'm,age of a pseu'd'o-comgtl'ote space. Proof. Krom [34] shows the existence of a metric Baire space -x such that XxX is not a Baire space. Suppose ihat there is a pseud.ocomplete space Y aniL a eonti:ruous d-open function f: Y-->x. Define n: TxY->XxXby n@,U):(f(n),f@)J for each n,!eY' Now YxY
is a pseud.o-complete space [52], anii hence, a Baire space. By Proposition 4.8, F is a continuous d-open tunction so that X xX would. be a Baire space. This contrad.iction completes our proof. Closed. continuous functions in general do not pleselve the Baire space property. Such an example is a closed map on the countable closedspace which is discussed in l4l (also see 1131, exercise 14, P. 253). Tliis space consists of {(a;,0)l re Q}u{(plq",Llq)l plq,Q} with the topology inherited from the lrsrr&I topology of the plane. The map is the natulal projection onto {(4,0)l ioe Q}. On the other hand, closed irred.ueible (i.e., the image of a prope closecl subset is a proper subset) continuous functions preselve Baire spaces [a]. In fact the following results are true. Trrsonnlvr 4.10. (i) Am 'irred,uci,ble closed funct'ion i's feebl'y ope+t' (ii) Tha i,rred,ucible closed comt'i,ttuous i,rnage of a Bai're space is a Ba'ire spa,ce.
(ii|) The elosed,
cont,imuous ,image oJ a paraconlpa,ct Ba'ire space
in
tho
strong semse 'is a paracornpact Ba'ire spaca 'in the strong sense' Proof. (i) Let u be a nonempty open subset of x. since / is irreducible, there exists ageY such that f-'(y) c [/. Since/is closed, f(X-Ul is closed in Y. But g e T -f (X - U) = /(U), so that / is feebly open' (ii) This part now foliows from part (i) and Theorem 4.1. (iii) The paracompactness is preserved. by closecl continuous functions a result in [37] 1481. Now let C be a closed. subset of Y. It follos's from that there exists a closed subset A otf-l(C) such that' f (A) : C andllA is irreducible. Therefore C is a Baire space' so that Y is Baire in the strong sense.
The last part of Theorem 4.10 is a generalization of corollary 1 in of a complete metric 16?l which says that every closed. continuous image space is a Baire space. The inverse image of a Baire space need not tre a Baire space even if the function is continuous, open, and" has each point inverse a Baire spa,ce. tr'or example, Krom [34] shows the existence of a metric Baire space X such tinak XxX is of first category. The projectionmap
Baire
48
.
spaces
from .X xX onto X is continuous, openr and has each point inverse a Baire space. The next two re:rults [21] illustrate the role playecl by the notion of eountable pseudo-base in connection with inverse images. Turonpu 4.11. Let f be a comt'inuous open' function from a spuce X w,ith a pgu,ntable pseud,o-base onto a space of second categorg y. If there is a, subset Z of T sueh that y - Z i,s of first category 'i,n T and f-r {z) i,s of second, categorg in itself fot' eaclt, ze Z, tlten X i's of second'.category. Proof. Suppose trlnat, X is of first category. Then there is a sequence {Enl tr,*:7,2,...} of closecl nowhere d-ense subsets of X such iha1,
: "
P:rr,,.
For each
l
.l
ext*b he'e*.
; S;*ry.
Fr#
sI,@ be *a ac{e
ra le,t
th** ibe $
M(I,): {uuy 1 intl_r1y;l.f-'(ilnn,f +o}. I-./et {al a,nd d let
i:\,2,...}
be a countable
pseud.o-base
* de*
for X. For each ra
: {yuYl g +J-'.(y)oUoc 3.n\. Since / is an open fu:rcbion, M! :f (Uo)-fLU$(X*8")l is a closecl set in f (U). Tf. in.tM| +6, then O 1f-r(intMi)nUac. -F,,. Ilowever, this would" contra&ict ?,, being nowhere d"ense in X since / is continuous. Thus, Mii is a nowhere d.ense subset of Y. Cleafly, M(I*) - f;,lf; to, ti:7 .each n.. Therefore, for each n, M(Fn) is of first category in Y. I-.iet ilI : l) M (F"). Then M is of first category in I.
,
ew.4-
,
sss*s
t&sw
xti:
@
n-l
-M) nZ. lTow is a closed" cover for n Since f-t(z). /-r(e) is tE*of-L(z)l =7,2,...) .of second. category in itself, there is a positive integer k snch that
$1.a.
.
proof.
LFk^f-r(z)l # O. Thus,
ze
r
,"1
f&e{F.
,.i , '.'' *!@ :
..
li,
{@
".ry #:l
Since Y is o{ second" category, there is a point ze (T
inty-r1ay
.
:
IL This contrad,iction completes the
Conor.r,Ea,-y 4.72. I'et f ba a conX,itr,tr,ous open fu,nction fro*r, a, spa,co X wi,th a cauntable psattd,a-base onto a Bai,re space Y. If there'i,s u subset Z of Y such tkat y - Z i,s of first category i,n T and, f-'(r) 'is a Baire space Jor each ze Z, then X is cr, Bai,re space.
ft is shown in [46] that the converses to Theorem 4.11 and Corollary 4.72 ate false. That is, an example is constructed- (nsing the continuum hypothesis or Martin's axiorn) of a separable metrizable Baire s:pace X which has a d.ecomposition into closed" subspaces each of whieh is of first category. The natural projeotion / onto the cluotient space Y formed" b1' this d.ecomposition turns ou.t to be open map. Therefore Y is a Baire space, but /-r(g) is of first category for every y in T.
#
*:
,:: '::ri
s{s ,
i
;,,;,1
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2. Baire
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9,t-$ i*r sJ
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space extensions
By using certain collections of open filters on a spacer lnany diffslsnS extensions of that space can tre obtained.. we will restrict our attention bere to what is normally called. the strict extension [9]' If -x is a topological space, an open filter base F on x is an open of X containing U, t'hen fiker on X if whenever A e F antl 7 is an subset : an opett u'ltraf iltet" is an a. if. be to said. be will free )g v e F. .Ltso F of all open filters. f:et J7 coilection in the open filtel which is maximal disjoint union of x the let x(3) be be any set of open filters oL x,'and : Uv{FeFl UcF}' Note and. -F. tr'or each set U, open in X, let U* that (u nv)* : uo ^v* for every open u aytd Y in x. T"iet z(J?) have the topology generated- by the trase {A*IU is open in X}' Now X is a dense subsPace of X(.nr). clearly, lt x ana Y aye homeomorphic and" -E and. G are the sets of open ultratilters on x and. Y, respectively, then x(7) and Y(G) are homeoriorphic. In fact, if / is an open continuous function from X into Y, then there is an open continuous function g from X@) into Y(G) such that
slx **ed
spaces
: f'
PnoposilroN 4.13.
If
there
u'e
lLo
free open u'ltrafi'lters on a space
x,
X i.s a Bai,ra sPaae. proof. Il x is not a Baire space, then there is a set u, open in x, '@ such that U: U-rYa, where for each i,, I{ac X"'+r and lfo is a nowhere g be an open dense subset of" u. For each i,, Iel aa : u -clu'-lra. Let ultra,filter on X containing the point finite open filter fuase {Uul 'i : L,2, '..\. Clearl;', I is free' fthe proof the next theorem (see [40] and" [35]) is omitted. since the proof of Theorem 4.15 uses a similar technique. IVe will say that a set tlr,en
of op"ofilters ? on a space X is
of open subsets of
ad,mi.si,btB
x with ut+r-
uo
if for every collectiott
and.
2or:6,
Ql
:
{Ur}Et t}ten there exists
such tbat Qt c -F. Aclmisible sets of filters include all openfill,erst all open uitrafilters, and- all free open ultrafilters' Tmonn r 4.7+. p,is an adrnisible set of open fi,lters on x, then, x(F) 'is a Bai,ro sgtuce (i,n fad X@) i's a-farsorabl'a)' Tsnonna,r 4-t5. If I i,s the set of all open' filters on the space x, then X(n) ds a Ba'ire space 'i'tt' tho stron'g sense' Proof. suppose that -E is a closecl subset of x(") that is of first category in itself. By Theorem 3.1J-, there exists a countable point finite coliection {vtl i,:L,2,...} of open stbsets of l7 which coYeIS l7 and is locally finite at no point of E.. Norv {t}* nE I u is open in x} is 4 base for the topoiogy oo p-. I-,et I/, tre open in X such t]1'at tllng - Vr.There
a^TeI
4
-
Dissertationes Matbematicae ct9*(fr) as follows. Let, f eg(g) and, V : (Vr,...,V,,)e 6*. ff there is a fi , & r1A) and, a V e fi suehthat( Zr, . . ., V *, V :lt, A, il.t ^*, ^+r) : Ynr+t; otherwise,tet yiliyd for sorne od.d z, then let yi,6)g) : i^". If !ler? T_ *- f.9r(4) and a V,,+t,g sudn that (Vr,...,V,o,vn+r) :lu'f ,filttor some even i, thenlet y?(f)(v):vm+ti otherrpise iet
* 9.
',|r
vb6g)
:
Y*.
Suppose that fr. is a Baire space. Let, U ,
gn
and,f
e
g"(g). By Theorem
fi.sqh) such that ff b(U), f (il,Alt *8. It can i:L beshownthat fl lU,f ,yrr(gl; t'fr.ThusXisaBaire space byTheorem i:r
3.16, there exists a 6
3.16.
Let 0,6 and. i, S1n1.Vy Theorem 3.16 there exists a fieg*(#) such tiiat fr b-r(fr), yrfri), gli +g. h can tre shown that ff La,i, fi@)lt+d]'nno*, -i is a Baire i-l space by Theorem 3.16. tr'or a topological space (x,{)r -f is metrizabre since it is a subspace Now suppose that
X is a Baire
space.
of Bairets zero-dimensional space 1491. Thus, by Theorem 5.21 and oxtoby,s exanrple d.iscussed above, it can seen be that there exists a metrizable Baire space whose square is not a Baire space. cohen [75] has recerrtly shown that only th: usual axioms of set theory are need. to prove tha existence cif a Bzr,ire space whose square is not a Baire spa:s. KLom's t:chnique above wili give such an example
which is a metric
sDace.
: ,,.' ir:.: 1",.
i'.r:'
a::,
F{#} :t
'
tba*
fu** Bibliography __.r )
i,saq :::i.
1'
j tsf
::*tlr .-:
;8?* t4Fi j:'&nd.
#. $t."ri+
:',fss" :ry+Z/
g,Xet ,i,.. tr?*.'3u ', ,:-
;i,'t€rl
j.
ry€r]l 1*
Br
$" gJ; rerTe :.
r:'. ,
arrtt R. I1. }IacDowe]r.l, Cotopology far m'etriaabl,e Duke Mathematical Journal 37 (1970)' pp' 291-295' ,tw1},-nta&ger spa,ces,Proc. [2] *{.arts, J. M., antl D. J. Lutzer, The ltrod,act of total,l' (1973), pp. f98-200 Amer. l\{ath. Soc. 38 _, *, psoud,o-contltletemess' aii tt, ltroilwct of Baire sp(rces, Pacific J. Math.
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[3]
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Lindgror.,A
Math. 24 (1973), PP. I86-187.
$
*et
ware iwple
f19l Fogelgren, J. R.,
ancl"
toytol,ogg,
North-I{oland Publishing Company'
n'ote om sf'anes of secomil' category' 'Lrah'
R. A- McCo!,
by Som'e toytol'ogdaal' ltropret'i'es d'efined
homeomorlfitisw grollJtr,s, Aroh. Math. 22 (I97L), pp' 528-533'
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$c**.r&g :
ia&- E;
", l:,'.
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i 1g*i5i, ,,,
.:3 6{}, :
&' Soc' gBlllgue
$tg**u'
::
.
.S,si-b. ..:..:
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g3s?4), t. a..
l
rgcces, I'r:'
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-.:,i
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*e$Fli, ffi-
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