POLSKA AKADEMIA NAUK, INSTYTUT MATEMATYCZNY
l;t,
DISSETATIONTS MATHtrMATICAtr IRO
ZPRAWY MATEMATYCZl\ KOMITET BEDAKCYJ
E}
NY
KAROL BORSUK redaktor ANDRZEJ BIAI,YNICKIBIRULA, 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. ftBaire 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 pseuilocomplete 4. The Banachl{azul Eame 5. CountabiyBaire spaces
,. :.1 , ':::
1i}
to Baire $paces 1. Baire spaces in the strong sense.
1. Blumberg type
:
11
l'oncepts relateci
Characteri.zations
;EF,
8
First and second oategorY sets Baire spaoes Isolatecl points and Baire spaces
44 4S
Bpaaes
53
56 I}()
60 o+
69 12
ir 'iri
i ;'
I g
i
Introduction
the Baire Category Theorem, versions of which were first
proved in 1897 aniL 1899 by Osgoocl [51] antL Baire l5l, respectively, &at every cornplete metric space is a Baire space (the term Baire I was introd.ucted. by Bourbaki). This theorem is one of the principle thlough which applications of completeness are made in analysis. ({completett space only 1e lr*" ,i*stances one needs to have a m such as the Baire Category Theorem, so that being a Baire lrilr,all that is really necessary. This is tho case in such weliknorrn as the Closecl Graph Theorem, the Open Mapping Theorem ancl Bound.edness Theorem. llhe Baire Category Theorem is useil to obtain certain topoiogical results'suc]r as establishing that ,*idimensional soparable metric space can be embeclde cl in Euclid
I
space.
Uhe tnown results concerning Baire spaces and relatecl topics aro in variety of papers and books. ft is the purpose of this tract anil organize most of these resu].ts. We have attempterl to inclithe various theorems have occured.. however a number of the t'heorems can be found. in a number of sources, so that we have a reference for every theorem. AIso some of the theorems are ,' have never occured. in a published paper before. ce letters N, Q, and. J will represent the natural numbers, the :sulnbersr and the irrational numbers, respectively. Euclidean eill be d.enoted by 8". If l" is a subset of the space X, then the cLosure of 4 relative to the space Z will be abbreviated as gLxA, respectively, or just int,4 ancl cll where there is no
*@
*$
$$
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 dense 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  clI[ 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 intclll :8, W intersects X c1rY. {ii) i,mpl,i,es (iii): By letting V : an(Xcll[), we have the clesirecl result. (iil) impli,es (i): If intcll[ *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 dense set of points, there exists a nonenpty open set r*ntajaed. in I/ which intersects only f initely many membels of {clM" I rt e A}, s*y {ci}I..}zl : 1 ,2,..., n}. Since each .XcllY". is a clense open sub'qet n tXc1]ro)] +6.8y the choice of 8f X, we can see that W :Vo[ i:I fi2 is open' W c XF, *.2(Xc1I{")  X [l clAr"' Since 1L
*qgt€$
XcI(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 7nli:o. Now there :6' ojren,in X, such tinat' V : Tf nY. Thus, W c U and' T{nN lsow suppose that Y is open in x ancl that M is nowhere dense 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 dense. I{otvever, this is not necessaril;' 111r" for infinite products 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
ir
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 cll[": 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<
jl
rvoulcl be sornewhere clense Ln.ll
X"'
ffSr some p eA, Np is nor.hele clense iu XBt then{Ji'" is nowJrere c fJint,cllro. Now suppose that clI'" **Il,:e in fJX" since intcl(IIf") ,irl ueA. : I, tor Lfinitely rnanv 4.,4, ancl 1et Lr ire any basic open sgbset of
Baire
a peA
spaces
that *t(A)
X* and*I*iu * Xr. Defiue a sutrset V of U by np(V):Xpcll[, and, o..Jrlr:*"{S;tor att other oe y'. Then V n fl N. : @. n X". Then therelexists
su.c'h
t
ry$
'tt"
*nS
s.ff S
!
2. First and second category sets A subset Y of a space X is of first crtegory (also called.
{n.
wb&€
in X if it is the union of a countable family of nowhere dense subsets of X. \Ye rvill say that T is of first category if Y is of first category in itself. A subset Y of a space .X is of second category (a1so called manmeager) in X if it is not of first category in .X. Tfe will say that r is of second category if Y is ef seconcl category in itself. Proposiiion 1.3 poirits out that if Y is an open or d.ense subset of a space X, and if 4 c Y, then the eategorr of .4 rela,tive to Y is the same as the categorv of 1 relative to X. PnoposruoN 1.5. Eaery den,se Gostftset of a, spdce of second, catagorg is of secand category.
ie j*
X be a space of seconct category and let O Uo ]re a clense of X. Now .X n Ur: l) (XU) is of first category in .x.
,.,,,i$
Proof. Grsubset
Thus,
T'et
i:1
Ut were of
t:l
filst categorl, then X
"9"
7": U X",r. For eachz,letllo : [J tl i:l in X by Propositiort I.2. Thus, say
r\ro,,..
Thenffjisnowhereclense
ue
l)et c. lct(L)o/)
U r'") : U /:lui aeA
that Nal is of filst category in X.
[cr(gw)
*l) rrl" ( i i,:I
is,..+
1..S,,.t
...,{
ur
d. "
*:i , &i
@
:
effgs#
;su :.1:'::
E'W: ' *{ s{tl l::,:.
".
woulcl be of first category.
Eitirer of the fo1lo'rving two theorems ([53], p. 62; [51]) can be referrecl to as the Banach Ca,tegorv Theorem. Tnpomlr 7.6, In a toytological space X, the qln'ion of aW fam,itg of open. sets of fi,rst category 'is of first categor'y. Proof. Let Ql be a family of nonempty open sets of first category. Iret B be the set of all pairwise clisjoint colleetions of nonempty open sets with the property tha't each member of each collection is contained in some member of a/. By Zortls lemma, S.has a maximal element t/'  {V"l aeA}. Thus, ct(g+tll)Nr is noN'here clense in X. For each O" represented as a countable union of nowhere d.ense sets, aeA, Vo
so
in
F
e
fi n i:1
rneager)
'.,1
.,.1
ti
I
as Rli
ft€j
t:i't
""' 15,&:;
@ ..
t,lr.3 rri.i::i
@
q
:
,,
{.8
Fe*
is$'#{
rqq,ts .,:.,: trr,)
It also follows frorn Proposition. 1.2 that the union of a farnily of first category subsets of X s'hich is locally finite at a d.ense set of points of X is of first categorJ' in X.
Try @ry
I*&t
',ry
l,l.l'.;.J
ffi
'il:'i
:;.
''
I.
Basic properties
.
T$eonsM I.7. Let A be a sttbset of the space X, and sttlrytose that for g xonetnpty oyten set U, there e*'ists a nonenxpty'open set V conta'ineil' :',srleh, that V nA i,s oJ fi'rst c*tegory dn X. Then A i,s of fi'rst category
, Froof. If / is nowhere d.ense in X we are through. So suppose that iintcta #fr. T'et' {Uel 0eB} be the famil;' of all open subsets of X contained in U and whose intersection with / is of first category Therefore, for each B<8, Aual is of first category in U, so that bf first category in cll, and. thus of first category in 4. By Theorem .$ {Cunl) is of first category in 4 ancl, therefore, of first category '&tB (.4 n U) , lf ow l) (U uaA) is a d.ense open subset' of An f/. Therefore' FeB in XtUEnA) is nowhere d.ense in /nU and', thus, nowhere dense
U is nowhere dense in X. Thus, 4 is of first category in X. 1.8. .4 sqtaca X 'is of second, category i,f and, on'ty i,J the 'inter${;rrl o/ any (monototto deuoasi'ttg) segtence of den'se opem subsets 'is non
, {' 'Treonpu
f i'*oot.
Suppose
that {Uo} is a sequence of dense open mrc
an empty intersection. Then
*f tiist category.
X : X.')
i:1
X
can be represented
@
*'sfo)  Xl) Io:6.
@
  m rr^* X : tJ "Fo. Now t:I Therefore, by Proposition 1.2, {XPr}
.,r.1 ^^r^^ r^^^ sets, r closed" ^^ v nowhere dense ^^a* say union of
:':.
X
Ur: U (.X  U). Thus, i:7
:.,,,lfow suppose t.inat X is of first category. Then ii:,,:;, :.?j.::. ,a.conntable
subsets of
fl
*:':*lonotone d.ecreasing seqlience of dense open sets with an empty lr:?noposnroN 1.9. Eaery Trspace w'ith no'isolated, poi,tr,ts haa'ing a olofinite base has a iletzse subspace whi'clt i's of Jirst categorll.
"Proof.
@
T'et
# : l)
Bo be
a olocally finite base for X with
Bu
o./i}. tr'or eachi, and ae 4,', let fiio, Uio and" let ff, : {rl,l ae A;}. X and. each'i, there exists an open set 7 containing r which xsects only firitely nranv tletnbers of Bt, sa.y a'"rr..., U!n. Now j : 1 , ...,1i there exists an open set TZ" containing r but not fu:.Gq{.h k ing'*rir. Therefore, 1t^(,V I/"r)is an open set containing n ancl conj:r than possibly r' fhus, ]Io has no limit other of no element l[, {.,tr1
'esrh +e
is closecl. that there exists a nonemptlr open set O eontained. itr ff,. n'tbere is a nonernpty open set 7 contained in U which intersects :.itriit"tl many rnembers of B,' Thus, 7 is a finite set which means and, so
$i*l ii:
lr:.:.
'
t0
that
Baile each point
spaces
of 7 is isolated. This contradiction yielils th*t
[] i[, is a d.ense subspace of I
]=u i.s
now
i
is *f
whicii js ot' first
arlJi €
category. To obtain a product theorem for first category, oxtobl' [b2] introducetl the notion of a localiy countable pseuclobase. A collection I of nonempty open sets in a space .X is a pseud,obase for X if every nonempty open subset of I contains at least one nenber of 9. A pseudobase I is saicl to bel,ocall,E countable if each rnemller of I contains only countably many members ol 4. Let A be a subset of ZxT. Tt n, X, then .4, will denote the set y u  {n, ll)e /.}, and if g e T, th.en Au will denote the set {n e Xl (n, 1y) e A}. {U Laltrra 7.t.0. Let X and, Y be spaees with Y haaing a countable Tiseudobase. If N ,is ,ttowltore dense (of fi,rst categor'y, resp.) i,n XxY, th,en N,'is nornhere d,ensa (of fi,rst category, resp.) i,nY for all, n encept u sot of fi,rst cntegorg in X. Pr o of . II'e will prove only the case where lI is nowhere d"ense in X x Y since the first category case woulcl immediately follow from this. Let {Ur} be a countable pseud.obase for Y, and tet G : (X x Y)clA' which is a dense open subset of X x/. X'or eae.h,i, Let Go: jlxl(Xxu)nflf.
€
hele d.ense in .X. Clearfy,
i:L
X.It )Gr:fr, t]nen X rvoulcl be of filst category by Theorem l.S; ancL *" i"iota be through. If re ) G.;, Thus, Gt is a d"ense open subset of
.
tlren one can easily
see
that G"nUo
open subset of Y. Thus, for
*
6 tor all z so that,
nehgo, TG*is i:t
i* p#, ** 3
ea€.h'
eralit
{:
ca'try prsE'*
aqd x a}l .{rii **o&
'j
*nbe
u:t;$
N
G*woulc1 be a clense
nowhere clense
c {## Se.e,*mt
in I. But
'l
4F** :'
4* tG,. fherefore, {ne Xl l/" is somewhere d.ense in Y} c  ),'*o " : y_r(" G1) which is of first category in X. Tnnonp1r a.LL. Let x amcl, Y be spaces wd,th at ,east one of tlr,ent, lrau,ing alooall,g countable pseudobaso. Let Ac. X anrl B c. T. Then AxB is of fi,rst categorg ,in, X xY il *nd only i,f A i,s of first categorly ,in X or B ,is of f,irsl category i,n L Proof. IVithout lorss of generality assume that Y has a locall5. cotntable pseud.obase9. Thus, eadn fJ, CI has a countable pseudobase. Assume tbat, A x B is of first category in 'f x y. If A is of fjrst categorl,' in ,f rve are throug'h, so suppose that 4 is of second. categorS' in X. tr'or each ti e P,
(Xxu)n(,4 xB) : Ax(B^U) is of first category in Xx I/. By tire I.,emma, there exists an r in /. such that [4x(Bou))r:BaA is of first category in f/ ancl therefore in I. Thus, by TheoremlT, Bis of firr,tcategoly in Y. Clearly, if B cloes not intersect l)9, tt.en B is nowhere clense in Y. Conversely, it is easil;' seen that if C is of first category in X, or if B
:*:
r
N
1!:
.
{,
h**
*
.:*
.
SSe&.l .:
gh.e#
ae{,:i
..e:
.3
,.
sn
lj.
4. 'E:,
,174
.,fiii
*t@ €ees*
fir.st category in Y, then axB is of first category in x x I, without 'restrictions on X or Y. ConOr,rl63y L.12. Let X and, Y ba spaces with ut laast one of t'hem h'aai'ng t,g cou,ntable ytsaud'obaso. Let Ac X and, B c T' Ih'eto AxB i's of iategorg i,n X xY i'f an'd, only i,f A i,s of sacond' categorg i'n X und B
categorg in Yf1.*g above theorem easily extend.s to prod,ucts of finitel)' many spaces' db..ot which has a locally countable pseud.obasel but, it does not gento inlinite prod.ucts, even when each space is second. countable.
second, : of 
Countability is an essentia,l part of the d'efinition of first and seconcl fcgorv. In ord.er to rid himself of this notion but stili retain the desirable oirti.*, de Groot [26] generalizecL the d.efinitions of nowhere clense rclsecond.categoryb)'definingforanycardi:ralrn'ran'rzthinsetancl ne,Baire *p**. r\sthin sets coincitLe with nowhere d.ense sets, and ,Baire coincides with second category'
3. Baire
L
spaces
Bai,re sltuce is a topological space such is of seconcl categorY.
that eve )/ nonempt;' open
oap r 1,.1'3. Ihe followi'tug are equ,'iaalent for a space X: (i) X 'is a Ba'ire lgace. ,intersection of any (monotone d,eereas,ing) sequo'tbce {11) The
T
sots 'is d'emse'in
of
dense
X.
of any sot of Jirst eategory i,n x is d,ense i,n x. (iv) EaerE countubl,o union of closod, sets wi,th no 'irrteri,or poi'nts i.tt x no'interior poitr't i.n X. proof. (I) im,plies (ii): suppose that {un} fu a sequence of tlense {nt)
Th,a comTtlement
sets,ancl '@
thatu isan opensubsetof xthat doesnotintersect )at.
u:u)uo: i:r
u(uuo),
nowhere dense in x. Therefore, u is of first category. a closed nowhere d.ense subset = tol im,pties (iii): For each ir let fr be tx, ancl let, U be a nonempty open set that does not intersect x[J rYa
u_aois
@
E
Vn: i:r )
(X.ffc)'Then {I/"} is a monfr tXXc). n'or *i' pffi d".tuasing sequence of dense open sets whose intersection is not =
each ra
define
I2
Baire
spaees
(iii) i'mplies (iv): Iiet L be the countable union of closecl interior points. ff 1 contained a,n interior point, then x,{ be dense in X. (iv) i,mpl'ies (i): suppose tbat for each z, ry', is nowhere
Tiith no rsould. not
!
se*s
6hs4.''
the*
$
h x,
s
and (Jlr; is open' Then each crr[, has no interior point, but rj d]ro d.oes have an interior point. 1; PnoposrrroN 1.14. EaerE olten subsgtace of a Bai,re sgtaee is a Ba,ire
llrOFd *i
d.ense
o
sp&,ce.
Proof. Let
x
let u be open in x. The conclusion is immecliate since an open subset of U is also open in f,. rn contrast, not eyery closed. subspace of a Baire space is a Baire space, as carl be seen by taking the space Er{(r,0)l ne{. The closecL subspace {(r, 0)l rrQ} is clearly of first category. Trlronpu r.75. Eaery spaca whi,eh conta,ins a dense Bai,re subsytace ,is a Bai,re spa,ce. Proof. Let Y be a d.ense sutrset of r, and suppose tbat x is not a Baire space. Then there_exists a nonempty open first category subset of x, say y. Thus, Y nu is a nonempty open first category subset of y. Pnoposnron 1.16. x'is a Bai,re space,i,f and, otr,ly i,f the comgtleme*
of
be a Baire space, and
nonempty st+bset of fdrst category i,n X i,s a Bai,ra iporr. P'oof. rLet a be a nonempty subset of first category in the Baire space Jf. By Theorem 1.13, x a is d_ense in x. rf l3 is of first category in xA, then B is of first category in x. Tirus, avB is of first "ate!.ory, in X so that .x (AvB) : (XA)B is den"qe in X_e. rf there are no nonempty sets of first categorv in x, then x is clearly a Baire space by clefinition. ff ,lI is a nonempty nowhere dense subset of x, then x N is a dense Baire sutrspace of x, making z a Baire space by Theorem 1.15. The next proposition ([dB], p. 4L) characberizes ail subsets of first category in a Baire space. eue41
Pnoposruon 1.17. Let, y be a subspace of the Ba,iro sgtace x. Then T is of .first category i,n x i,f and, onry i,f xT conta'ins a d,enieGysubset of x. Proof. suppose that {lrr} is a seq'ence of nowhere clense subsets
of ,x such tirrat, Y:oVUto. Since
Z is a Baire space, ..i,tclxllz) is a clense G6subset of X contained in Xy. Now suppose that fr *,, t* a d.ense Grsubset of x contained in x y€ i:7 ?hen fl (XG) is of first category in .X anct contains y. i:L
Pnoposmron 1'18. rn a togtorogi,car, spaco of open Bai,ye subspaces i,s a Ba,,ire spa,ce.
x,
tho union of any fami,ry
sp*@,
E
df E#
*
*ad&
tr r.j
bat&
"
IS e&
Bair*
Bu'ii
g
se.
,,:,1...8
rs*.I :.4.
p"
:.:.,
aa$&
isaS { 13
4/
*€* #
i:E
IF
ds*S
'"lEi
€:
Ls
aS #
f;"a3s
*g,
"g
tbee',& the, r.*i be *,;3
ro*&*'. a 6*'t€
;ft s:?sry1,
l.
iE
ID
be a family of open Baire subspaces of X. Suppose first category subset ot" l)a/.ILet U be a member ot Ql intersects I/. Then U nY is open and" of first category in U.
Proof.
a* l&€, e*t ryg&fu
C
Basic properties
F is an
TLet alt
open
, Clearly, the arbitrary union of Baire subspaces need not be a Baire si.nce a singleton set is always a Baire space. But as the following l:i#** ' proposition states, it is true for finite unions. r.,
tr.
sry..
$ *{3j il,
*r.@* #!g#.&;3
*,@
iry ]..:,,'
:dsp*se
*.F**e ,S :f,
Proof. \Yithout loss of generality'we can let X:TvZ where Y end Z areBaire spaces. It is now sufficient to show tlnat X is a Baire space. XclY and. Xcl Z are open in Z and. Y, respectively, and. hence are ;..futh Baire spaces. By Proposition 1.18 B : (X clY)v(XclZ) : X(cl TacIZ) '.;.,,, is an open Baire subspace of X. Since XclB is an open subset of the
. I
',.
AuiXclB)
is a d.ense Baire subspace of .X. Actually Proposition 1.19 is true for a locaily finite collection of Baire
'u***
PnoposntoN 1.20. Eaery
ii&@
Proof. Let X :
*&ffi'
*q+q
,S
"1,o ""be
di,sjoi,nt
topologi,cal sum
of Ba,ire spa,cas
the d"isjoint topological sum of Baire spaces,
*rt
S*w :F
#4rFe"y:**
;
**.
esj"$ :l :,'., ,
i
S'..,F,
i"
ry
fi'
''
'kw.a neighborhood, whi,ch 'is u Ba'ire sp{r,ce. : .._ IYe have alread.y notecL that if a sp&ce contains a clense Baire subspace, ',,' then it itself is a Baire space. The rationals consid"ered. as a subspace of ,. , * *berea,ls point out that a d.ense subspace of a Baire space need not aiways , , .be a Baire space. We will now discuss some ad"d.ed. cond.itions that will 'i : '" 'naske a d.ense subspace of a Baire space a Baire space. The notion of '." * #.subset will be fundamental in this discussion. ,:' ltoposrtron 1.23. Euery d,ense G6stcbspace of a Bai're space'is a Ba'ire .1,,,,.:.
1A
Baire
*paces
Proof. Let G be a dense Grsubspace of the Baire spa,ee 'E. By proposition 7.17, xG is of first category in x. Thus, G is a Baire space by Proposition 1.16. Parts of the. next theorem appeared in [3]. TunopBr11 7.24. Let X be a dense subspace of the Baire space T. (i) X r,s a Ba,ire space if and, o,nly ,if eaery som.ewh,ere d,ense Gosttbset of T inl,ersects X. (ii) x is a Baire space 'if and onLy i,f eaery Gosubset of r corttuitrcd,
i,n Y

TX
,is nowhara d,ense i,n Y. (iii) I/ euery Gosubset af Y contuiyted, X, then X 'is a Baire sp&ce.
(iv)
ff yX,is
ersery Gusu,bset
of Y
dense,in Y, then contuined,
i,n
YX
i,n
TX,is
mowhere tlettse
X'is a Ba'ire space if i,s noc,uhere d,e,nse'in
and, onty
yX.
in
if
Proof. (i) Suppose lhat H: A O, is 'a somewhere dense Grsubset of r contained in yx. Let Y il'on",' in Y and eontained in cirB. Then VnZnU, is open in VnX, fr VnXnUt:6, and VnX is open i:1 in X. To see t]nat VnXnUo is dense in I.rnX, Iet, neVnX, ancl let U be an open subset ot v nx that contains :r. Then there exists a set G.
in r,
u:Gnvnx.
,
Gnv contains r and is open in Y. Thus, Gnvnun is a nonempty open subset of r. since x is dense in Y, GnVnaonX is nonempty. Thus, UnVnAonX is nonempty, which gives us t'hat vnuonx is dense in ]rnx. Tirerefor.eo x coulcl not open
such that'
Now
be a Baire space. Now suppose that z is not a Baire space. Then there exists a set 8, open in x, and a sequence {uo} of clense open subsets of E such flrab
O Ue : A. Let T/o and K be open in Y such that (J, : V,nX and" E : KnX. F,ac]n Zu is open and dense in the Baire space K. Therefore, n T/o is d,ense fu K. Thn'qr ) vo is a somewhere d.ense G6subset of y ir i:r contained. in YX. To see that (ii), (iii), and (iv) are true, consider the following three statements: (1) Every somewhere d,ense Gusubset of Y intersects X. (2) Every G6subset of Y containecl in yX is nowhere d.ense in Y. (3) Every Grsubset of I contained in yX is nowhere d"ense in yX. The desired" conclusions are immediate when we realize that with just 2t containecl in Y, we can get that (1) is equivalent to (2), (B) implies (1), and, with the added, hypothesis tltat YX is d.ense in y, we get that (1) is equivalent to (3).
E
*",
*
ryry
dsl
.:
:,**li * ,
s, 1*{ {re
r
.:ii,*,.
:1;.'
..
s E:iI,i
alFialt:
'
r.
r..ri;:
Basic
properties
15
r  ^^^:!r^^r f from ,;,,..",,,r;i"=*,,...Ehe hypothesis tha't YX is clense in f cannot be omitted' {irj of the above theorem. This can be seen by consid"ering Y to be re&ls ancl X to be the reais with the nurnber 1 cleleted'' trIie no.w' wish to characterize spaces of seconcl categoly in terms of .i{ai.iir,.j .t t
i# tt',: ir:+.t*i:,i*f .
r#{mr.;rrr€ t,
tlt
he'er aet;i
4 i:
,,
spaces. . r ,:i:.e.'
pe"oposrrroN 1.25. In eoerg topotogical sp&ce X there are open (possibl'y) *.,".,.,i.,"'i;. ' : a rt. f iegFyr subspaces Xu and X" srt'clt' tlt'at ,' 'ti XunXo:bt and' XuvX' i's dense in' X; : ', Ir",,:r . {ii) Xo i,s a Ba'ire sqaco) liii I artd X, is of fi,rst categot'Y' proof. Itet Xs be the union of aII open first'category subsets of X. .t,. .,t
&*
i,re3u:et ;,
Trreonnu 1.26. A sqtace x i,s of second, category i'f and only i,f it co'n'ta'i+1's e rianenpty oltett, Bai're subspuce' proof. If x is of second category, then xr in Proposition 1.25 is &flEenlpty. conversely, if x contains an open Baire subspace, it mu'st'
i
*:3t&. .",1
:. *gog!:
:, ..
i?*e f
:
l'.!d *. S eTl,f*
tre
of
seconc[ eategor)'.
Though the concepts of spaces of second category ancl Baire spaces
*.:.d*xane
$', there exists an g
*Sli{F, .g,e+€'g. '.
vr&
.gL
.!" '
pnoposrrrox 1.27. Eaery "Bct'irely"
'
*+at
sptv'P
.
a
',ga:d.
:; rd T
1
.
I
*'ie
n'
: I3Yali
'e'it&
.:
11
be any open sub:et of
z.
Then, there exists an #e
uI
Berre.iubspace of tr. Thus, f (U)nlr is a nonempt)'leconcl category subset {it l rliricir gives us tirat T/ is of second category i1 f,' Thus, in such \ homeogeneous space is "Bairel;"' homeogeneous. i?il.ffr1le.i as topological groups, Baire space.c. and spaces of second category Rre er.luiralent concepts. In connection with this Bourbalii ([13]' p' 257) sr*.tes that a colntable topological grotlp v'hich is a llausclorff Bajre space l* fll:CT€IP.
,
f*rg.t"
A
honteogeneotts second, categoTy
is a Baire s\a(e'
5 .*: I:. Iret
s
*rrxfe:.e*.
*a
X such hhatf (n)< V
,$
1
' *!
in U ancl aflnction/trom X into
,.
:,:,.,
4.' Isolated points and Baire spac€s
,
The existence of isolatecl points pla5rs 2 funclamental role in the study
il.lwlFe*
r:s.t.*ffi ;...
;,,,::.,,..,+
.,C'h'. r'r.r*.
16
Baire. spaces
FnoposrfioN 1.28.
Xp
In oaery toTtologi,eat space X tlwe
are
ogten.
{possibly
g'
Xp suclz thai (i) XunXo:frt and, XuvXy,'is dense i,n X; (ii) .xB is a Bai,re space; (iii) and, eaery si,ngl,eto+t subsot of Xp is tr,ornltere d,ense ,in, Xu. Eurthermora, X is a Baire space i,f and, only +f Xu i.s a Baire spece. Proof. Let B : {ne Xl intcl{r} +g}. If n< A c B, flren g fintcl{n} c intcl4. Thus, every nonernpt}' subset of B is .qomewhere dense in x. Therefore, eyery nonempty subset of 3 is somewhe.re clense in 3. This means that B must be a Baire space. Consequentlr., intclB is a Baire space. The first part of the proposition now follows by cle,fining Xn: intclB and. Xu  XclB. The last statement follows from Propempty) subspaces
and,
i*
{fffi:
:,.
5,**e ,_
r'
.&ss*1
E
*md&
X.
s**&8{$
Proof. Suppose
'_s
G: h
is a countable somewhere d.ense ", Gosubset of Z. Then there exists l,loou*nr, open set U such that U c clG. Now every singleton subsetof xis nowhere dense inx. Thus, Gnu is of that,
t; _
: (cnu)uI u (Gnu]l : (Gnu)u(u  fl ar) :
Therefore, u is of first category in
x.
Gnr)"lLt (J 
i:i i:r This, however, contraclicts
x
ilF
#;S;
{d gR
first category in X. tr'or each 'irIet Uo : VtAU. Now (A A)nG : 0. Thus, since uuacclG, u c'; contains no open subset of. x. To see that, U*Uo is nowhere dense in Xrlet V be any open subset of .X. If ynU :4, we are through. If. Vn{J #6, then W :(V^U)nUo #0 since UUu constains no open subset of X. Now W is open in _f,, W cV, ancl T{n( O  r) : O. Thus , U Oo is nowhere dense in X. No.ry U 
a Baire
a._1
?b*rc
ositions 1.14, 1.18, and Theorem 1.15. Pnoposrrrou 1.29. Let X be a Ba,ire Trspace wi,th no isolated, pto'ints, and let G be a countable Gusu,bset of x. Then G ,is notnlrere d"ense ,i,n
&,q
*i 5..s
.E$. EE
u)1. being
space.
Pnoposrrron 1.30. rf x i,s a countablo Buire Trspace, then, the set of isol'ated' poi,nts of x i,s d,ense in x. rurthermore,,if x ,is coutztabty i,nfi,ni,te, then the set of i,sol,ated, gtoints of X i,s ulso 'infi,nite.
Proof. Every nonempty open subset of" X is a countable Baire frspace, and", therefore, by Proposition I.29, contains an isoiatecl point. Suppose that X is countably infinite, and let ,4 be the set of all isolated points of. X. ff ,4 is finite tb,en XA would_ be open, and, therefore. contain an isolated. point. PnopostrroN 1.31. Let X be a Baire Trspace ai,th, no ,isalated, pto,imtrs, and' let G be a somewhere d,ense Gosubset of x. rf 0 is a cou,ntable subset of G, then GC is a somewhere dense Gosubset af X. 2
.&i!
', ;:rri
:.
l
'.
ii
I. :' Proof. NowG
a countable
: f\ i:l
suhset
1t7
Basic properties
AtisaGusubsst of
of G. Thus; G
C
:
XanilC : {rtl'i:t,2,...} 6 O (Ao{n}) is a Gusubset
X.
Suppose tt:at G  C is nowhere d.ense in X. Since G is some*hore in X, there exists a nonempty open set U such that U c. cl&. Let
V
: UnlXcl(GC)1.
that V is nonempty since GC is nowhere depse .*1(7nG), so that VnG is somewhere dense in X. Also
in X. Now Iz
VnG : anlXol(G C)lnG  [X{GC)]nG : C. trnG is a countabie somewhere dense G5subset of X. This, , is a contradiction to Proposition 1'29.
PnoposruoN 1.32. Let X bo a Ba'ire Ttspace ui'th no i'solated' Ttoi'nts,t Iet G be a sotnemhere d'ensa Gusttbset of X. If D i's a d'ense fi'rst categorg
of X, than G n(X  D) 'is wn'coantabla. Proof. Suppose that Gn(XD) is countable. By Proposition 1'31, 'l1en(,Xr)l :Gn,D is a somewhere dense Gosutrset of X. Thus inD is somewhere d.ense in ,. Now X  D is a dense Bairo subspace Xby tneorem 1.13 anil Propo.sition 1.16. This, however, is a contraclition
t, $'
il F.
Theorem 1.24. :
$r.
Dissertationes Mathematicae C:
.
rI." Concepts related to Baire
spaces
In the filst three sections of tlris chapter we introcluce and. rliscuss sereral classes of topological spaees that conlain the Baire spaces. Each of th.ese classes will be of importance in later clevelopment. of course,
the rnain result is the Baire Categor;' Theorem which is the fouaclation of the original interest in Baire spa*u*. The last section of this cha,pter is concernecl with the impact that the notion of a Baire space has on minimal ?t (resp. rrausclorff, rrrysohn, regurar rrausclorff) spaces. rn conclusion \ye see that everv Baireclosecl space is tinite.
l.
Baire spaees rn the ,tuong
"*ni.
The concept of Baire space in the strong sense ([20]; [13], p. ZbZ; [28], p. 165) is probably the most natural concept to introduce of all those which imply the concept of Bair.e space. A space x is a Baite space i,,tr, tlze strong serzse (aIso callecl totall,y nonmeugei') if every nonempty closed subspace is of second. category in itself. The next proposition clearlv inclicates flrat every Baire space in the strong sense is a Baire space. The example given after proposition 1.14 is a metric Baire spa,ce that is not a Baire r pace in the strting sense. Pnoposrrrox 2.1. x i,s a Ba,ire sltaee 'in, the strong sense i,f and, only i,f eaery nonentTtty closed, su,bsqtace ,;s i Baire spa,ce, Proof. suppose ura,t x is a Baire space in the strong sense. Let l? be a none{npty closecl subspace of x, ancl let u he a nonenrpty open "subspace of F. clearly,3 is a Baile space in the strong sense. Therefore, clp' [/ i.r of second. category in itse]f. Now U is a d"ense open subset of crv[l. Thus, by Proposition 1.5, u is of seconcl category. conversery, every Baire space is of second category in itself. Pnoposnros 2.2. Etsery Gusu,bspace of a Bai,re space ,in, tlte str"ong
X i,s a Bai,re space ,[,tz th,e strong sense. Proof. I,et G be a nonemptS'Grsubspace of .T, and. let ? be a non_ empty closed sutrset of G. It can easil;' be seen that n: Gnclr_F. Therefore, ?.is a d,ense Gusubspace of the Baire space clr?. By propsense
osition 1.23, I is a Baire space.
ws
ffi
,
.,..
:
l.i
IL
Concepts relatecl
to Baire spaces
2. Baire Category
It'
Theorem
ii'i''
I:ocall}' compact Ifausclorff spaces anal completely metrizable spaces important classes of topological spaces wirich are of a totally rwo i;*re ; 'r1:' .:':. dfrferent nature. One reason for the abstraction ancl inclependent studv I i,,.,"=.".. *f Baire spaces is that botir of these classes are contained" in the class of ,' ' I Baue sJraces ([]31, p. 193). Tirponnlt 2.3. Eaerg locally co'mltact Eazt'sd,orJf space X 'is a Ba'ire .'.,. ,:_ ... l_ anil ltence i,s a Baire space'itt, the strong selLse. ' r1r*re. ttl' of Oense sequence oI dense open SubSetS subsets oI of X, ancllet L', De a Sequence Ul Ploof. Let IDit Pr'oOL ,., ,' . {r andlet {Dr} be .,.:.i,:..: t'. ' is of .X. Since X localiy and regular, cornpact nonernptlr open subset :lt.t . tre ativ nonemptlr "ovexists a nonemptS' set Ul open in X, whose closure is compact '. +here ,''i l'. . ' a:rr[. eontained in TiroDr. In this manne 'we can define, for each i,] 7, .'E uouernpty set Ua, open in .X, wliose closure is compact ancl contained:., . Norr {cUol i>2\ is a decreasing sequence of nonemptSr .l:. ;.. , ..t in C 'ADi'. , rlsserl iubsets in the compact space cIUr. Therefore, )cltlo 4 A. . .
*ircus,q
Earh
'
€r*T:Iie,
*+.l;rtion *4?Ailter ''&:&i O11 e.*;19+
fn
p. 9;13;
* *f ail
I
l
@€erl:l
f is uow a Baire space since (l clfiu : (\tit  U'n (n ,r).
The T{ausd,orff hypothesis in Theorern 2.3 cannot be omittecl since ' ; ' ,1., , * *ouutably infinite set 'with the finite complement topology is a compact .. ?rsperce which is of first categor;'. Also the converse of Theorem 2.3 . l i. f:rlse as .El with the lowerlimit topology d.emonstrates. ' ' a Brtire sp&ce, a'n'd h,ertne i,s a Ba'ire space 'in th,e sh'ott'g selxse. I Proof. ILet, {Do} be a sequence of d.ense open subsets of X, ancl let U, ,a ' tre rlonempt)' open subset of X. Since X is a rnetric space, flrfiD, cont;:il. i1 closed. ba,Il TI, of raclius r'r.In this manne \Ye can d.efine, for each :. .
$lg t**r;
s
**,*,elf.
a:
ii:.
r?B
lke
314
it*rda titli;{ }.'
'o6@
f
. .tili:,' J:.
if
\ !, . :
t1FeI1
< r, < r;r12. Thus, {Aii'>2} is a d.ecreasing sequence ot nonempty closecl *ets whose cliarneters converge to zero. ft is well linown that in a complete .,,,.., ;r:etlic space, such a sequence has a nonernpty intersection. X is now rr). & g,i1i1s space since O Ilt: 1l int {Iuc (}na .{t
{n
6if#S*ril.
r.
f
..;.tr
g
&*kr
:, ,' ,r r*€S{riqf
li{* X1{.1{}
':'i.*g.r*

I r?rri*
,: I
.
Yan Doren [6?] has shown that the closed continuous inage of a comgrletr rnetric space contains a dense comp1etel1. metrizable subspace. g'fur,*. the Baire Ca,tegory Theorem is valicl for every closed continuous ia:rlge of a complete metric space (see Theorein 4'10). the esample given after Proposition 1.14 is a metrizable Baire space qhss ir neither topologically complete nor locall5r compact.
g.
imply Baire E\, e nill now clefine some cotnpleteness properties that have been stud.i{14 in errrrnection with Baire spaces (see 14] for a stucly of these properties). Qernplete type properties which
Baire
spaces
A Tyghsr.ff space x is complete 'in the sense of dech if there eriists a sequence {4to} of open coyerings of x such that for every famill of ciosecl sets {F"l aeA} which has the finite intersection property, and which lras the property that for each ,i,, there exists an Fo contained. in some fieQl;;
)F"+6.
if;; i g.E:if
'{
aeA
rt is weli known that for a metric space complete in the sense of decir is equivalent to being topologicall)' complete. There are many equivalent definitions of spaces that are compiete in the sense of (iech including, foi: example, a Gu in any rlausilorff compactification. The above clefinition [17, p. 143] was chosen so that one could. easily see that the dech complete spaces contain the almost countably complete spaces 1201. A space is quasiregular tf every nonempty open set contains the closure of sorne nonempty open set. A quasiregular space X is almost countably cotnyilete if there exists a sequence {9*} ot pseud,obases for X ,such that for every sequence of sets {Qt,,r} wbidn iras the finite intersection propert}
ldl,**.
ffi*:E d*4.*
EAJti r
{r,
iri€
ir* i
fl,oeT"ni 1clU,r+9. A space X is pseu,clocotttgilete if it is quasiregular and. if there exists a sequence {3} of pseudobases for x snch that for every sequence of sets {I/,'} with Uirgi ancl cl Ut+r= U, for each z;
s€s:
2r,*r. Pseud.ocomplete spaces were introduced in
.
and.
It:
I
lbzl after.which an irr clepth stud.y of the concept was d.one in [3]. clearly, every almost countably cornplete space is pseud.ocornplete. Aarts and Lutzer l3l show that a metrizable space, in fact a Mosre space, is pseud.ocomplete if ancl onl.v if it contains a clense completely metrizable subspace. Pseud"ocompieteness has an interesting generalization called weakr'11 afarsorable (originally defined in [?0]) which utilizes ideas from game theory. The definition is given in the following paragraph, ancl a discussion of the relatecl concept of afavorable can be found in i16l (also see the section on the BanachMazur game). I'et (X,f) be a topological space, and let {* : {Urgl U +fr). (x , g) is taeakly afauorable if there exists a sequence {/';} of functions such that the following hold: (i) For
earh,
'i, ilre range
of.
f o is
d#., :'k
t:/ ll
:i.
3&.!'t;,
s;&
t.
i*#
*e***ag :,sgl
*; :.,
,#g.'+,t
tr$r**
{*;
(ii) The domain of /, is g" and. fr(U)  U for each (J e{*1 (iii) tr'or ezch,i, the domain of /o*, is
4$e}e
tt,
d@i t
{(Uy Ur, ..., Uiar)l U1,9* and. ap,  f rc(Ur, ...., UN) for all i:L,...,d*1 and" Jt,:!,...,i);
,ta
dtr g.$,
ffi@ ,t ,:l!ir.r' I . .:,
,,
i'
$
t1.
is
,
i,
11. Concepts relatetl
f ,'(Ut,
i**recl
'",
a
r*r) c
A
;.{*I1l€
' ,,,
Letrll
l:ient *!!9, $tian p.}ete
Irl ;inl!]'e
i*illt!l
thet i.*rt.r 3l1$rii
U;+i)
i+i
(v) If {I/r} is a sequence i'€cr each i, then A U; *6.
'i;!rh
such
is in
2L
the d.omain
that (Ur'...,
Uo) is
of fi+t, then
in the d.omain
of
/t
C:t
that every pseuclocomplete space is x'eakl;' afavor, .*,$1e. \Yhite [?0] shows t]rat the concepts of weakly afavorabie ancl p*euc1ocomplete coincid.e for the class of quasiregular spaces whicir have metrizable subsPaces' ,,,: *€nse I Tnnonnn 2.5. Eaery weahlE afuuorabl,e spa,ces x i,s a .Bai're space. :: Proof. supposethat x is not a Baire space. Then thele exists a nona countable . enrptl' open set U that' can be representecl as the union of m Io' Leh {fu} fumilv of elosecl nowhere dense subsets of X, szr'y U : U i:1 r,' }e the sequence of functions gualanteeci by the definition of weakly ','l',sfayorable. We construct the sequence {Ui} of open sets as fo}lows' '''jf,*et Ut: U antlr assuming that U,, has beeri d'efined, let arr+,
',, :
rr,",h
if (U1,'..,
(iv) For arry 'i)
igists
to Baile spaces
i;
It, is easy to
yn1Ur,, S@
) i:l
u
i
.,.,
:
Urr)
U'n
see
_Frr. Now
ft(ut, .'., In i:r
Uo)

E,]
*{l i:t i{al:t
'' This, however, is a contrad.iction to part (v) of the definition of weakly , atarorable. ,. The example given after Proposition 1.14 is metrizable and contains iA I dense completely metrizable subspace' Therefore, it is pseud'ocomplete ,. *nd., thus, weakly ofavorable. Ilowever, it is not a Baire space in the ,,.: ,g6,s1g sense. The next theorem uSeS a construction d.ue to BernStein
r?iei a es:rtr5
*lj}*€3r
rec3!e rii*.ASi;
p. 514), and gives us a wa\. to construct a Baire space in the strong ,. ,Lrellse that is not weakly ofavorable. ' TqnoREn 2.6. If (X,{),is a septarabl,e conrpletely metri,zabla space wi'tk t" p* isolated, ytoints, then there an'ists a subset Z of Xwi,ththe followi'ng prop
\ga
41361,
=,@_3.
t*E"
d
. : e$*es:
;..'
: t,:
:
'
li) Both Z and' XZ aTe d'anse'i't'L' X, haue card'i'n'ali'ty c, and' are Bu'be bpa#s in the strong sense; point) tii) z/ T i,s a su,bsgtaaa of z or x  z th,at d,oes not haae an 'isolated, fSaru f is ttot weaA'Ly afuaorable. Proof. It is easy to see that lXl :lgl: c' Thus' each Gdsubset sf J without any isolated points has cardinality c, and there are exactly e r:f them. Lrct {n"l a < c} ancl {D" I a < a} be well orclerings of x and the gslleetion of Glsubsets of X without any isolated points, respectiYely.
We shali use transfinite induction to d,efine {p"l o 1 c} = {n,! a < c} ancl {q"l o
D"
and" goe ," (u\fue, P, &0, e,.j
ge.}"
{p,})
6@
.
i:l$
e&K.
Let, y6 be the first orclinal such t\tat mroe Do, and, define ?o : fiyo. let do be the first orclinal such t'hat nuoe Ji and fiao * ?to, a..d tlefine let and suppose T4ct and {pel
3
r, (p {?e, 4e}u{pd}) for errery 6 < y. Let yr.be the fiJJuo"clinat B<6
Dr Ur{g o, qu} (such an element exists since the carclinality of U {pe, q* is less than e). Define ?y: nyy. Let Ay be the first orclinal such that nou, Dr(p^,{ge,g,,}_{pr}), ancl define gy: n5,. Ltet Z:{p"L o
{Ftr#
i., ,e ,':l .
::*.g*w
iF;'= a.: ..
ET.',i
is nowhere dense in F. Let H : etxF ancl Hu : cl,Fifor each .d. N"orv E is a completely metrizable G6subset of x that does not contain an clense
in 8. Therefore,
:: rr'l
I
I i7
E, is nowhere
.q
,6qF
rt z is not a Baire space in the strong sense then there exists a set F, closed inZ,w]nicinis of first eategory in itseif. Thus, F : _Fo wher.e ?,.
isolatecl point and eacb'
5;
:ge*l
S, ....
e,*
*.siF
n. () go
:.O (lTEr) is a dense Grsubset of H. Th's ) E3 Uris a GrsFiset i t x:l of x that cloes not contain an isolated. point. since F is closed in z. HU Euc. Hn c. X2. This, however, is inconsistent with the con_ i:1 struction af x2. similarly, xz is a Baire space in the strong sense. suppose that Y is a 'iveakly afavorable subspace of z or x z tbnt does not contain an isolatecl point. Then firere
$eEkl
@
er:ists a aompletely metrizable dense subspace of u, say D, which, in this case, cloes not contain an isolated point. D is a Gusubset of x since the complet'el5' p"1"rable subsets of x are precisely the G5subsets of x. This, however, is inconsistent with the construction of Z or X2. The countable intersection of dense Gosubsets of a Baire space is a d.ense G6subset and a clense G6subspace of a Baire iipace ii a Baire space; this suggests consid.ering countable intersections of dense Baire subspaces of a Baire space. rlowever, with a proof similar to that of this previous theorem, a monotone sequence of dense Baire subspaces may be constructecl in any separable completel5' metrizable space with no isolatecl points in such a way that its intersection is empr;,.
Y34 ]J
ffi_L:
,
'1eq. lr r
,:.,. i i *;
;,;.:&&*e ES.s
,.,sf*u ..:... I
t.'
*fi*si
e"*
"ry =fl #f* ..
,.;.'.
E€PF
t
t,
:'
.. ,
II.
ffi
Concepts relatetl
to Baire
spaccs
,.,. \ye now give a rnore general r.elsion of the Baire Categoly Theorern ,i:*hen statecl in fheorem 2.4. tr'irst rve neecl some clefinitions. , A ytseud,o=sernimett'ic for a topological spa,ce (X, {) is a function d ,:tn XXX to the nonnegative reals such that fol all points r ancl y of 'f
et
,1.
d *ny subset A of X, .,',: {1) tl(n,Y) :0 it r :'li ,'., (ll) d'(n, A) : d(Y, n); i,,'.i (iii) and u.d.A if and ",'d
'
i*
kfine *r:ld" "..
,
oJ.;i
on15
if d.(n,A) .:inf{d(n,a)11 a<.4} :0.
..g,space (X,{)isacompleteTtseud,osem,itn,etricspaceif thereexistsapseucLo:,,&,mimetric for (X,.n such that eYery Cauchy sequence in X converges
:\i.]i}al
fo a point in X'
: ,,
lirsi
: '.agtuce
T'V &zr
#t, F;
tt'li.'5
ip *n *
{X,.7)
'is psetrdocompflete.
..',:: Proof. I'et' d, be a pseudosetnimetric for (X,f)'tr'or r'I , ,a) 0 let
" L€t
i,t1*:
PnopOSrrrox 2.7. Epery conryflete pseu'dosern'imetric quas'iregular
i:
ancl
:Kow { For each i' 1et {tl(u,7li,)l n<X ancl i :I,2,...) is a base totgoit a pseucloirase ,Et. {u(n, e)l nex ancl 0 < e < 1la}. clearly, each ,,f*r f . I.,et {t}i@,t, e1)) be a lequence of sets rvith Uo(no, el)r 9, ttncl ,,xlliiat(#*r, u.i+r) c. (In(n;, er) for each i. Now {rol i' :1,2,...} is a Caupoint !/, X Thus, ,. *lrr sequence in X ancl, therefore, co\erges to a .S6
,, di e+3:
uaQs;*t
..
ir, 7
i.: i 4:{F:Lt*gC*e.
r
rk*:
k+€6q si, xbe; 1,. xYE**e
EAriil$:e
g$
&iegel
i.?ia+*eei
l:"
i3i${}g*5
1$r.'€*{!
*w4gs
:
;l
Two other related properties, cocompactness 11] ancl colntable
sab.ompactness [26] hacl their origins in a,n anall'gig of the Baire Category lsheore. The clefinition of cocompacl,ness uses closeil bases whereas the ,,, ,*e.finition of countable subcompactness uses regular filter bases. Both : {l6ncepts are equivalent to completeness in rnetrizable s]laces. ' A closed basa for a space (x,{) is a famil5' fi of c'losed subsets of 'tx,{\ such that for each point rcx ancl each set ur{ containing r there exists some Brfi;;ucin that;e intB c.Bc.cIU. x together witll I €.he topolog;' generated }:y the ,qubbase {X B i B e fi}t will ]:e called,x. cosp&ce of (x,{). A space rs co^onxpact if it' has a compact cospace.
. emptS open sutrsets of X such that whenever U and. 7 are rnembers of '2, *a"" tft""" exists a member W of F with lTr c U alr.9 isregtr,lar" if, wheneFer t is a mem]:er of F, t]nen there exists a 11e111ber 7 of F wibh' clY c f. A space (x ,t) is cortnlably sttbcontpact it tirere exists an open base # fol (x,g) such that whener.er {un} is a countab}e regular filter base tl o + g. **ntairred" in 0,
Q
Baire
24
spaces.
Tneonnm 2.8. Eoerg qu,as'iregula,r conntably subcom,pact sgtaae i,s a Ba,ire sp&ce.
Proof.
X
I
be the open base for.ll guaranteed b)'the d.efinition of countably subcompact, {D} a sequenco of dense open subsets of X and O1 any nonempty open subset of X. Since tr is quasiregular, there exists a nonempty set Ur< I such that clUrc UroDr. fn this m&nnex.we can d.efine, for each i)1, a nonemptj set Uae @ witin clUnc. Uo_rnDo_r. Now { u1  i > 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 rlausdorff 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 quasiregular almost countably complete spaces, the pseudocornplete spaces, the weahly crfavorable spaces, the quasiregular cocompact spaces, and the quasiregular 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 llausdorff, rnetric, completely normal llausdorff, locally compact lfausd.orff, or zerod.imensional llausd.orff. tr'or an in depth stucly of minimal topologtcal spaces see [g]. Given a topological property P, a Pspace (X,f) is mi,ni,mal P i!., und.er the partiat orclering 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"
.
.. iltJC€.
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 ?rspace that is of first category. The following proposition gives a criterion lor cletermining when a minimal ?rspace is a Baire space' PnoposnroN 2.10, A m'ininzal, Trspace (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 : Xl) 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 io 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.,(UIf,,)l woulcl i:r be of first categorv in (X,9). PnoposrtroN 2.\2. Eaer31 m,ilti,mal (7, Ba'ire)space is fi,ni'te ot' /tL?I,
then,
cotctr,table.
Irct (X,,,r') be a countably infinite Baire ?rspace. 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 ?rspace "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 Trspace.
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 ?rspace. Thus, f :f* "since ({,/) is a minima'r (7, Baire)space. conversely, if (x,g) is a Baire rninimal ?rspace, it is surely a minimal (?, Baire)space. There are theorems for rfausdorff, 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 does 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. $'eaker topology on certai:r kincls of spaces. PnoposrrroN 2.14. Let (x,f) ba a Eausd,orff (resp. arysoltn, regu,lar" Eau'sd'orff) Ba'ire spa,ce, and, let F be a norlcow)ergent open (resqt. (Irysohtt,, regular) fdlter base wi,tlt' a unique adherent poi,nt 7t. rf g*: where // : {Uef I ptU} antl "tr : {UvVl TteUe{ and, VeF},"//vJr, th,en {* ,is a Hau'sd'orff (resp. urysolm,, regular H*u,sd,orff) Rai,re topotogy on x tha.t i,s
properly {. Proof. {* is a Baire topology on x by proposition 2.11. To coinplete the proof refer to l8l ancl lb9l. Pnoposrrrox 2.15. Let (x,t) bo a regular Hau,sd,orff Bai,re space, q'ncl let F be a nonconaergant, open (rospt. urgsohn, regul,ar) Ji,ttey base wi,th a u,n'iqu,e adh,erent poi,nt p. If g :.1/v{, wh,ere .,V7 : {U
t'
Fr.il
Fe, es,?s
*,freei
esdu
I t
i i$
.a,, . ,t,
di$4
$
f** &&g '.1 ..:.
*Iialg
l
$Es{
if
*:
i*x
3*gs
i.i
SSt.Sil
.:
*s*&.* +l
:
SI}H as
*}a
t
': tmffi $s}ry
t.s t f**r#
..]
T€.*
L 4 *:{
&?Stt€q,,1 ,i.
{*R*
II.
Concepts lelated.
to Baire
spaces
@r .:,
U is a member of the topology on X generated by the subbase {X  cl U l U e 0\. This topology is now a Baire topology by Proposition 2.11. To complete the Proof see 1681. The proofs of the next two theorems are omitted, since the;' e,1u similar to the proof of Theorem 2'19. Tnnonnu 2.1,6. (X,g) i's a mi'ni,mal' (Hausd'orff Baire)spa'ce i,f an'd, only i,f (X,g) i,s a Bai,re mi'ni'mal' Euusd'orff spa'ce. Tirrronovr 2.1?. (X,f ) 'is a m'ict;imal' (Urgsoir'* Bai're)space i,f attd' only if (X,9) is a Baire m'inimul, Urysolt'n sf)s'ce. PnoposttroN 2.18. Eaerg mi,wimal, (regular Hausd'orff)sp&ce (X,9) is countably sztbcomPact. Proof. I.,et {I/;} be a countable regular filter base on (X,{) Berti and" Songenfrey l10l have shown that in a minimal (r'egular Elausd.orff)
#'$lf
spase, every regular
kat
Since {U;} is a reg'u1ar fiiter base, h Ao +A. d:7 innonpm 2.19. (X,g) is a'mi,ni,mal' (regular Hausclorff Ba,i,re)space i,f and, o'nlg1 if (X,g)'is a mi'nimal' (regular Hau'srlorff)spa'ce' Proof. suppose t]na,t (x,r) in a regular llausclorff Baire space that is not minimal (regular llausd.orff). Then there exists a nonconYergent regular filter base on (x,{) that has a unique cluster point. By Proposition 2.74, there exists a regular Ilausclorff Baire topology on X properly contained. in.T. Conversely, If (X).9) is a minimal (regular Ilausclorff)space, then it is a minimal (regular llausclorff Baire)space by Proposition 2.18 ancl Theorem 2.8. Thi: next proposition tells us that one needs only to check a particular class of Baire spaces to d.etermine the minimality of certain spaces. Pnoposrtron 2.20. If (X,.q) 'is a regular Eausd,orff Bai're spa,ce, lhen the following are equ'iaalent: (i) (Xrg) 'is a rn'i'tt'imal, Eau,sdorff (resp. []rysohn, regular Haasdorff)'
s:"fsrr
".
HtEC
*,i*:i
i;s: imea*l
Mft
w I**
33,1
t,*t $*&
flq5 *#i
sp&ce.
M wg* !#,F w$d,
${rs ,lr:
l. ,
::
€*#
l
filter
base has an adherent
point. Thus,
L cla; r
A.
(i1) Let @ be any o,penbase for{ and,Iat {* be the toptologSl ott x generatad, by the subbase {XclUl Ue g}. If {* 'is a Eausd,orff ('esp. Urgsoh'n, regular Eaztsd,orJf) Bai,re togtology, thett',f* :7. Proot. suppose t]nat (x r7) is not a minimal Ilausd.orff (resp. urysohn, regular lrausd"orff) space. Then there exists a nonconYergent open (Iesp' Grysohn, regular) filter base on (X,f ) that has a unique cluster point' Proposition 2.15 now gives us the method. of construction necessary to eee that (ii) irnplies (i). To see that (i) implies (ii) see [68]. An investigation of minimal spaces cannot be cond"ucted. "without eonsiclering Pclosed. spaces for sorne topological propertl P. lVe wiii
t ..
j
Baire
spaces
finite. If some separation property is tacked. onto Baire, then results similar to those produced in our minimal investigation would. appear. Given a topological property P, a Pspace {X,9) is ?closed if its image is closed. in every Pspace in which it can be embed.ded. PnoposmroN 2.21. Everg Ba'ireclosed, space 'is fi,ni,te. Proof. Suppose tbat (X,f) is a finite Baire space and. that p is a, point not in X. TLet, X*:Xu{p}, and let 9* :{U cX8l anXef, and if p" fJ, then .X* U is finite). (X,g) is a Baire space try Theorem 1.15, btnt (X,{) is clearly not a closed. sutrspace of 1X* , 9*7. see
that eyery
Baireciosed. space is
.,
ii
'{,
t. i..
l
i:.
i _11
':I
I
,'.
;,r
tt
{Le ir t$m*l
i*s
Af
III.
ffiint tr* ii ::,. &*.5.
Characterizations
of Baire
spaces
present certain interesting characterizations of Baire spaces. Among these will be characterizations centered" around. special types of functions, a covering characterization and a filter characterization. We discuss the BanachMazul game which leniLs itseU to a characteriaation of Baire spaces in terms of pseutlobases. In conclusion we introd.uce and. investigate the notion of a countablyBaire space which is'equivalent to the concept of Baire space for spaces having the countable chain condition. Charac,terizations of Baire spaces in linear topological $paces can be founcl in l5?1, [58], and. [66].
fn this chapter we will
l.
Blumberg type theorerns
Blumberg [11] showecl that for every real valued function / defined on the real line O1, there exists a d.ense subset D of. EI such that "f l.a is continuous. We will say that space X kas Blumberg's property with, respect to Y if for every function f : X>Y, 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 White's result [71] that every semimetrizable 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 discrete 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,osemi,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 pseudosenirnetric 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 ft (G"). To see that tr, is of first category in X let An be the set of all points rrft (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 fr(G)An is of first category that, are not heavy points relative to f1(G,). Now .B,, is a nowhere d.ense snbset of X. Ilence, 8,, is of first category in X since 8,,  Anu Bn.
Thus, XZ is of first category in X since XZ  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) : ft[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) <712n. ft) a/. : {U(2, e,,(z}l ze Zn).
fs
*s
*
i i
seconcl
I'et Zr: {rt U(2, er(z))rQtrl and for each zeZ,Iet Vr(z) : Iyr(z)a au\2, er(z))oz.lf ne ulz, er(z)), andBis anyneigh'borhooclof rcontainecl in U(a, er(z)), then BnTIrr(a) is of second. category in .X. Slnce X_Z is of first category in x, Bavr(z) is of seconcl categorr in x. Trenee, frr(a) is dense in LI(2, ur(z)).T,tet {'r: {Vr(z)l ze Zr}. T_tet Zo:fr, and. {s : x. Proceeding by induction suppose that the finite sequences {2,,i n  1,...,k}, {t,"1 n  1,...,k}, {q/*l n 7,...,k}, and {7.*l ,L : L, .. ., k\ have been so definecl that the following are true for each 'tt
&
i&
:
{
&
,
s,
.r:*
i€
4*
li P*
#'"u,
14
6;
i:t 1..
IlL
il' '5:.",.
cf Baire
:11
spaces
.,;
(vi) Ut*is Pairwise clisjoint' (n, \)q/". is dense in X. (vrr') {'* : {V^{a)l ae Zn} such that
Wv
l,**
$.. r'..
Characterizatious
'"X
IJ (z
S€d
SS" i,w.g.y
!ffpry itet a sffistfusaot
:h
i
:in
.E
* 4o
dtd" Sense
$rx;
fix i*
, e,,(z)\ oW *(z) . (ix) For each 2e
(x)
Znt \)f
rye
*t
ze
Zn,
ze Vn(z)
c
2,,, Zo(a) is dense in (Iiz, t,('))' n.
For eaclr a e Zptheteexists a 6*(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
Iret
'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))
'
Qlor@)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
:
{Tznr(ir)
%t
'
 ueZ1,*r{z)),
{**r:
g lwo*r{a}vl(Io*r(2, +t: seZ4
I
e*r(z))}]'.
therefore have ind.uctivell' definecl 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 fjle)ot,"l ??'€^7)' Since X.is *" frroo the.t (l ll)qr; ri,. l/fis dense ilX, 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,osemi,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,osent'imetri,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 pseudobase 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 od,i,sjoint psaudobase, 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. Itet g : ()
,1.
:r
,',
:.]:
,w.$
{*,,1 m<l[] be a pseudobase for ltl, where aach 9n is a disjoint famiiy, and. let Mo : g. By Zorn's lernrna there exists a maximal pairwise disjoint collection //, of elements in e s.ucih that' 9r  flr. Suppose that for each n,: 1r . .., k a family tt,,]nas been so defined that the following are true: (i) .t//,, is a pairwise rlisjoint family of open sets; (iL) U//" is dense in X;
*ss
some element, of ur'ln. each MTre//a let Mf+t  {.Il^npo*rl pt+rrgr,+r}. Again by Zorn's lemma there exists a maximal pailwise d.isjoint collection eernl of elenrents in I such that MI*'  QQI,) and" [J (Mn) = trI*. Let .//r*, U QQIil.8y the principle of finite induction we have a seouence
*xd
N:T
(iii) "//n refines jln_ri (iv) each Pne 9n contains
l.or
lI
w t.
Et
8S*
:.
e]t
p<. // 1,
{fr")L,
such that for each izetr7 properties (i) through (iv) listed above
are true, an.d
.// :
U_
IF; g€'
fl" is a pseuclobase for X.
3a
,
III. 6
IretZ:) }rret .l/(Z) : {lIoZl
s+le
I
ESr
! g'.
,f is a Baire space' Z is clense in X.
Since
to Y.
i'.
white gives the following example of a Tychonoffr cocompact, pseud.oBlumbelg's complete space (anil hence Baire space) which d.oes not have measllr.e Lebesgue p denote T'et reals [?1]. property with respect to the if [/ only TJe{ if and. follows. io n', and. clefine a topology { on 81 as re U, each is a lrebesgue measurable subset of frL, and for
otT FRevl
ws{jt
i ,rg,elfS i
i
ter, 0< p(r). +] *1 :limsup pQ) lrrru,closeclinterval, ;;; t[_jLa?ll  "t
**&1e Fx'Ee
'
l*'o*{or#
'
*$$d 'ij.. .
t,
kre
L
r'i' !.8{,}.
*ami
is
n,
closed,
interval, * e I, 0 < l.I(I) a ll' nJ'
is possible to eliminate all hypotheses on the d'omain in a Blumberg type theorem by restricting the range. The foilowing theorem suDrmarizes
:
kere
f
It
Mfle ..1
spaccs
in(Z,Z).ByTheoremS2(2,7)hasBlumberg'spropertywithrespect Thus X has Blumberg's property with respect to I'
a$b
,t!,.
of Baire
topology on X) and' 1et V (Z) M€ //}. L,et ,z lr be the relative topology on Z. Tber "// (Z) is a pseudo base for { (Z) on Z ' Now .t// {Z)is a base for a topolog.v 7 on Z. Flor each L1. i//, nf oZ is both open and. closed in {2,7). Ilence, (2, il is a regnlal space with orliscrete base ..// (Z). Thus, \Z , il is pseud'ometrizable' Xo.\r (Z ,{ 121\ is a Baire space since its complement is. of first category in x. Therefore, (2, y) is a Baire space since a. set is d"ense in (Z ,{ {Z)) if an6 onl;r i1 it is clense
i&s,t
:'
Cbaracteri.zations
:j
tl
t
','fuF
i'#6i
some of the results ([71]' [39]) along this line' Trrnoaniu 3.6. The .fottowi,ng statements nre egu'ioalent for a toltological space X: (i) .X is a Baire space. (ii) ry Y ,i,s a coumtable sltace and f ; x>Y, th,enthare'is a d,ense subset D oJ X such that Jlp is cont'inuous. (iii) ry f: X>EL and' lf(X)l{l{o, than' tlzere i,s a d'ense subsoL D of X such thut f lo is cont'inwous. (iv) I/ f i,s a functi,on from X i'nto the 't1'at'ural' mumbers, the'n thare 'is a d,ense subset D of X such, tkat f lp ds cont'inttous' (v) I/ Y i,s a seqtarabl,a motrio sqtaca and' f: X>Y, then for eaerg me D(e)' e ) 0, tneie ts u d,ense sttbset D(e) of X s'tt'clt' tltat for each
&*.
int{diam{/lqE(u))l u is an open set in,x containing r} <e
mec.e
*qive
iia
.
TT_ Proof. (l) i,m'qtti,as (ii): Iret U bo any open sutrset of X' Now point there exists a U{Unlt(U)lU, Y}. Since .x is a Baire space,dense in x' r,et Bbe : (y) is somewhere tl g. i such t'nat, x(a)
^f'
I  Dissertation3s &Sathematicae CXLI
r,i.:.]
,:i
.i
the set of all pairwise disjoint subfamilies of {UnintcIK(U)l U is open in x), and consider S as partially ordered by set inclusion. 3y Zorn's Iemrna, S has a.maximal element Qt. Let /' : {K(U)nintclK(U)lUn nintclK(U)ealj ancl let D:U/. Norv D is dense in X and K(U) nintclJ((U): DnUnintclK(U) for each member of 7.. Therefore, each member of Ir is an open set in D. Clearly, / restricted to any member of Iz is continuous. Thus, _f lz is continuous. (iv) i,mpties (v): I,et 4 be a countable base for Y such that diam(B) { e for each Be9. Pat' the cliscrete topology on fr and. define g: X># so that f (n)e g(n) for each ne X. By our hypothesis. there is a d.ense subset D(e) of X such that g laro is continuous. ft is easy to see that D(e) is our d.esired set.
(v)
i,rrytl,i,es
(i): Let {ffi} be a countable pairwise rlisjoint family of X, and. let lI be the nonnegative integers with
nowhere d.ense subsets of
Z into l[ as foilows: let /(r) :i it neN4, and let f(n) :0 if n<X i +. By hypothesis, there d:1
the trivial metric. Define a function from
D of. X such that for eat:b ne D inf{diam{/1"(U))l U is an open set in X which contains #}<+.
is a dense subset
Therefore, .f la is continuous. Norv DnNo : A fot every i. fhus, Lj t, i:l is not a open subset of X. The following theorem due to Schaerf [60] illustrates a strong generalization of Blumberg's Theorem in other directions. The concepb of a proper fi'rst coantnble space is usecl in the theorem. such a space is one where eaclr point r has a countable base {B*{r)} such th.at B,,*r{n)nBn*r(U) t'6 implies that weBn(y). Tmonnm 3.7. Let X be u propzr f,irst oountnble Baira spa,ce, let {W*} and, {7"} bo soquences of socond, countabl,a spa,les) and, Let {gn: W*>X} and' {f": x>Tn} be seqtr.ences of funct'ions. rhen, there er'ists a d,ense subspace D of X such that
(i)
each
f*lo is conti,nttous,
and
(ii) for each open subset U of Wn, g"(U) ,is optett, 'i,n D. Blumberg type theorems may arise while stud,ying special types of functions. This has been the case for ccontinuous functions [22], semicontinuous functions ([16], pp. 111, 114), ancl functions for which the inverse image of open sets are Jf" ([16], p. 110). L,evy l38l has introduced the notion of strongly nonBiumberg spaces and investigated. its relation with Baire spaces. An interesting strengthened. form of Blumberg's Theorem can be found. in [15]. lTeiss [69] has recentl;' found. a compact rlausd.orff space N'hich does not have Blumbergts property with respect to
the
reals'
i\
i.,, :.r,ir .1.
i
,i
:',.. , r,,,,
'
i',
@
ry
:w
III.
,fur:rts
#b,:,En
:,1{{r)
tr€rsre, Sa*aber
Wer{B)
",,549 Lfu our
oi #€:with
f{r)
there
.t.
d'L
's o'U irr' ,.,
t.l=t
'Wge''
@"t&e *:S
t$ro**ig)
{]tri
%*xi @*p{&ca .,
!:':i,,,, .:,.'
@i'.*fpes
ec tr
F991.
Tfffl
sffiErc€0.
' &tion #*
Ehe
@iBa*t
,'1
*1lt"r'
to '
,is
aBa,ire sllace
if
F"bot. o]fandlet/treanylowersemicontinuousfunctiond.e{inedcinU.For r}. since / is lower semieach rational number r Let G,: {ne ul f (n) > in I/' If r is a point of [/ continuous, each cIryG,G' is nowhere dense p > 0 such for which J is not cintinuous, then there exists a real number for,any *hat VnG, #@ for any rational number r'(f(n),f(a)*p)and be must there open set Iz eontaining r' Since U is of second' category' some point of U for which / is continuous' Nowsupposethat.IisnotaBairespace.Thenthereexistsanopen s"t u whicS^ia' be represented as the countableunion of nowhere dense inf{zl r'Nt}' snbsets of X, say U: tJ to' For each neU Iet f(u):
r,xtrbset
es
35
und, ong iJ for eact' (bou,nded) sam'i,of conti,nui,tu of f is d"ansoi'n x' points conl,,itnrows functio+r,f onx,the setof s_ubset suppose that x is a Baire space. I'et tl be any oPen
TrnonnmS.q.x
::
15, .
of Baire spaces
functions' Baire [6] was the first to d.efine antl stud.y semicontinuous properties using spaces Baire we will now give two characterizations of of sernicontinuous functions [20]' A real valued. function / on a topological space x is called LLppera tt for each real number sem,,icontinuous (retp. lower sem,icont'inuous) is open in X' a}) the set {n,Xl f(r)
ry *)Fen
Eei
Characterizations
,
Theng:tl"lfisabounded'lowersemicontinuousfunctiononU vn x, g carr easily which is continuous at no point of u. since u is open on x' be extended to a bounded lower semiconJinuous 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
oaclt, Jor "open sctbset
x,tcaneasilylreextended.toalowersemi.continuousfunctiononX. It should" be noted. that results similar to fheorem 3'8 and' Theorem ',: 3.9 holtL for spaces of second category'
Baire
spacos
2. Covering anil filter charaeterizations The concept of semicontinuous functions rvill now be
to
#,$
employed.
ee;
i,$i
a covering eharacterizdtion of Baire spaces (1401, [18]). Trrnonnm 3.10. Ilte foll,owi,ng nra egu,iaal,ant for a, spaca X:
estakrlish
(i) X zs a Ba,ire spa,ce. (il) Eoerg poi,nt fi,ni,te open
coaer
of
X
**{
€
i,s locallg fi,nite
nt a
dense set
4.w
i,s Local,Ly J,inite at
ryr
i,mpli,es (ii): Suppose that X is a Baire space and.Iet 6 point finite open co\rer of X. tr'or each r e X,let f (u) be the cardinality be a of the set consisting of all members of € wbicn contain r. Now / is a lower semicontinuous function defined. on X since, for each real number o,
*#
of poi,nts.
(ili) Eaery
aowntable poi.nl
fi,nite open coaer of
a, densa set of ptoi,nts.
X
Proof. (i)
{ne Xl f (a) > 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 quasiregular 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 quasireguiar 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 riith empty intersection is nowhere cLense (see [4?])' r,*g.rs. &el$iaa
at S*ire
raa"*
3.
of Baire spaces involving pseuilocomplete 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'iregula'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,docomplete; (iii) and, oaerE pseud,ocompilete 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.ocomplete subspnces of X and let Xn  XclX". Conor,r,mn 3.I4. Let
com,yi,ete sflaces. Then
X
X
be
a qu,asiregular
cou'n'table tr'mion of psetttloif X i,s a pseud'occtntplete
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 quasiregular Baire space@which is the countable union of pseudocomplete spaces, say X, : p, Xn.If there is an'i such that intclXt *@, then intclXn is
g
1:
Ii
=
{:: .{J'
ii:;
pseud.ocomplete since clXo is pseud"ocomplete. This contradicts part (iii) of
Proposition 3.13. Therefore, intclX n: fr for eachn.Ilencc fi tt,  e1,T,,) is an empty dense 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.ocompiete extensions. PnoposrnoN 3.15. Let X be a guas'iregal,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. $\ nor eaery pseu,d,ocomplete space Y suclt, tlrat X i,s d,ense in, Y, each Gusubset of Y which, is conta'ined in Y X'is nowhere d,ense,i,tl, T X. (iii) 7or some psendocomplete space Y such, th,at X,is dense itt Y, each,Gosnbset of Y wh,i.clt, 'is co'nta'ined'in Y X i,s nowhere d,ense,in y.f. Proof. For any pseud.ocomplete space Y such t1a.at X is d"ense in Y, the set yX is d,ense in Y, since X contains no open pseud.ocomplete subspaces. The proposition now follows immediatel;' from Theoretn
h:
g
nd
?t
.E
ir
*I,
1t.i1, 0,1
::t
4. The BanachMazur
game
Arouncl 1928, the Polish rnathematician, S. llazur inventecl a rnatirematical game nolv known as the BanaehMa"z'tu' ga,nxe [53]. Originalll' this game was to be played" on a closecl interval in the real line. The cLe
III.
Ej. H+ff. &,a€$
'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 wal 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 ofavorable if and. onl;' if player (X) has a winning strategy for the game G(X, O) rvith (CI) playing first. Thus, being afavorable 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 pseudobase for a space
X, Iet g(g):{f: 991 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 , gfr)) 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 aTtseudobctse 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'obase 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 pseudobase 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
pseutlocouer
,in X. (v) Let I of
X
:6
*x
I
: U lr. lYellorder 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,,iu) :0. Uence, fl lU,f ,glt r':i it .f,
w
be any psettdobaso for X. Tlten, any poi,nt fdni,te pseud,ocoaer 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 pseuclobase for f. Then there exists a membel a of I wirich can be expressecl aq 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 pseudobase
9(9). Brt znfclmi.tt, of ord,er n rve shall rnean a nested seqr.ience Gr  G, = ... = Gr," of 2rr, sets belonging to I srLch 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, sav 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 order ri has been so clefinecl that the set of all 2rz terms of elements of In is pairsise clisjoint and. their union is a dense 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 bV 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 pseuclocover 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 pseuclobase for j and let ,2/ be a point finite pseudocover for X containecl in Q which is not locaily finite at ant 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'ntablgBa'ire'space i{ there point finite pseudoexists a pseud.ob 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."Br* Theorem 3.16, eyer]. Baire space is a countablyBaire 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 bioprocluct 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" denotes 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 ltoprod'uct ]'s6 *p*Ju, are continuous ancl open, it follows lhat' X is of first categor5r' intercountabie all taking 0 clenote the pseuclobase 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 countal: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 defined' by: z"(P)
fi"\P):Q"foran}r0>B.Ilence,Pmustbedisjointfrornevervmembel
10
Baire
spaces
of q/) so th.at al could not be a pseud.ocover of x. Thus, g does not contain a countable pseud"ocover, so that x is a countablyBaire space. Without adclitional properties there is no relationship betrveen the notions of counJablyBaire 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 countablyBaire space. The above example also shows that the irnage of a countablyBaire space und.er a continlols open Inap need" not be a countablyBaire 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.obase 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 countablyBaire space. lYe rvill now investigate when the concepts of Baire spaces ancl countablyBaire 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,clobase for x conta,ins a cortn,table
pseud,ocoaer, 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 pseuclobase for x, ancl every pseud"ocover 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,dobase el'isjoint pseudocoz,er of th,e space.
# for a space cotttcr,i,n.s a prtir,u:ise
Proof. Let' I be a pseudobase for the space x. rrct .,// 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.ocover 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;;."<€i
ffir1 ::::
$s;e +€{:i1
w*d.
i,fP] i.. ,*re
;Flr :.: .,
....'
I E
i
Characberizations
of Baire
spacoe
/,
PnoposruoN 3.19. The followi'ng a're equ'iDa'le'n't: (iJ X has the cou,n'table cha'i'n condit'ion' (i)Eaeryltseud,obasoforXcontainsacountablaTtseud'o.couerforX. pa'i'rwi,se d'i,sjoitt't (111) Eaerg ltseud,obase for x conta'ins a cou,ntable pseud'ocouer for X. Trrnonpu 3.20. x ,is a countublgBai,re spaca i,J and, only i'f at I'oust one of the fot'lorni'ng two proporti,es hol'd'z
(i) X is a Baire spu'ce. (ii) X d,oes not haae the coumtable
chai'n cond'i'ti'on'
Since the propert)' of having the countable chain conciition proof of (iv) is an open hereditary f"op*"ty, a slight mod.ification of the space' ConBaire g'fO is a rrouta show that 'X implies (ii) in theorem 6y Propcondition, veisely, if X cLoes not have the counta,ble chain satisfiecl' osition 3.17, the d.efinition of countablyBaire space is vacuously The next propositions follow from Theorem 3'20' PnoposrrroN 3'21. Eaerg d"i,sjoi,trt topologi,cal, sum of countablll.Ba,it'e
?roof'
spaces 'is
a
cou,ntabl'gBai're sPacl'
i,s a cogrztablgBaire sgtuce wlt'iclt is tt'ot a Ba'ire i,s a coun'tablyBa'ir'e space. sq)(rco, th,en atsery dense subsgtace of.
PnoposrrroN 3.22.
If x
x
PnoposrTrol{ 3,23. Eaery sltace uh'ich conta'ins a d"ense cou'ntablgBaire subsltace'is a cou,ntablg Baire spa'ce' For more information concerning the relation of the countable chain condition to Baire spaces and the effect of replacing the continuum hypothesis by Martin's axiom, see [63]' we end" this chapter by giving an application of Theorem 3.10, which can be found. in [63]. Tmonuu 3.24. Eaerg ltoi,nt finite open coael' of a Ba'i,re space h'au'itt'g the countable chai'n cand,i,ti'on'is countable' Proo{. Ltet xbe a Baire space having the countatrle chainqlcondition, is locally 3.70, and Iet ?/ be.a point finite open coYel of x. B;r Theorem open ancl : finite at a dense set of points. Thus 9Z {u I u is nonemptjr pseudobase in x and intersects only finitely rnany members of 4t\ is a for x. By Proposition 3.18, I contains a pairwise d.isjoint pseudocover g, of x. since x has the countable chain condition, 4' is countable' member Now each member of 4/ intersects some member of I' and' each countmust be Ql t]nat' of 9' intersects only finitely rnah;' members of 4/ ' so able. cott'n'table Conor,r,my 3.25. flaery meta,comylact Bai,t,e space hao,ing the cha'in, comd,i'tion i,s a Lind,elif spa'ce'
IV. The dynamies of Baire spaces fn the first part of this chapter we cliscuss different types of functions tha,t preserve Baire spaces and give sufficient conclitions for the inverse image of a Baire space to be a Baire space. fhen our attention turns to certain filter extensions of spa,ces. tr'or example, it will be shown that every topological space is a dense subspace of some compact Baire space in the strong sense. The last section contains a cliscussion of the behavior of Baire spaces under the formation of hyperspaces and function spaces. 1. fmages and inverse images of Baire
spaces
\lrhen stutlying a basic topological concept it is ahvaS,s of interest to known rvhat types of functions preserve that property. \r'e will now see that Baire spaces are preseryecl by both almost continuous feebly open surjections [21] and feeb]y continuous dopen surjections. T'et f be a function from a space x into a space y. Function / will be called alnt'ost cont'inuous if, for every open subset v of, y, irI, c clint/.(o). Also / will be called feebly continuous if for opuo "oury subset Tz of I with/r(v) +9, there is a nonempty open subset u oi x such tlrat u  f'(7). tr'inally, / will ha feebly oTtetr, i$ for every nonempty open subset u of x, there is a nonempty open subset v ot r such that Y = f (tr). A feeble h,omeomor'Tthism is a feebly continuous feebly open
bijec,tion.
rt is easy to see that a function is feebly contiuous' if and. only if preserves dense subsets. Also a function is feebly open if ancl only if the fnr'erse image of every d.ense subset is dense. Every almost continuous it
iunction is feebl5' continuous. Tlrsonnm 4.1 rf f is an almost continu,ou,s feebty open fu,ncti,on frorn a Ba,ire space X o+tlo a space T, thett, Y ,is a Bai,re spnce. Proof. T'et, {va} tre a sequence of d.ense open subsets of y, and for each i let Un: int/l(Zo). Since / is feebly open, each fr(vt) is dense in x. since / is almost continuous, % is dense in f1(v1), ancl, therefore, is d.ense in x. rlence, Q uois d.ense in x since x is a Baire space. Again,
;
€
'*lffi
W
lr/. Tho dylamics ol
Baii'e spaccs
ot f, f (i Or) is clense in Y' Thus' O by the almost continuity dFi:l
f (?.Ut' ?r,u' Conor,r,my 4,2. TtL,e image of a Bai,re space under a cont,inttoxes Ju,neti,oat is a Bai're sPece. is dense
X
in f
Vt
since
PnoposrtroN 4.3. i,s a Bai're spa'ce.
If X
i's
o,pen
spuce, trcn' foebly h'omeomorpl'i,c to a Ba,ire
Proof.Theinverseo{afeebiehomeomorphismisafeeblelronreo.
S**ns se'rs9
ax to 3,5ery
*h* 1rI
;sgtol ic*{eE,
kvest i asw eebly
f :.r: 'will **{t} {3Ser}
:*f E tt:$t\I €hat
*per
'
.
qb if :S
the
R&SAS ,.,
. #*tlt 3&
ror
,'$ xse
re{ore,
@ia,
morphism, and every feeble homeornorphism is almost continuous' Y' Then / wili be Let, f be a function from a spaco .X into a spa'ce (]t) is a nowhere callecl ogtenif for every nowhere clense subset l[ of Y, ft ilense sirbset 4 every.somewhere for dense subset of X, or Lquivalently Y' ot X, f (A) is a somewhere dense subset of one of th'e PnoposruoN 4.4. A functi'on f : X">T 'is 6'open' if an'y Jollowi+r,g 'is true: of X (i) / r;s cont'intt'ous and, tho 'i'mage oJ oach nonemptg oyten subset ,i,s somezultere d,ense 'in Y,
(ii) / r;s con't'inuous and, feeblE opan, (iii) / i's a feobla lt'omeomorph'ism' Proof. Assume that (i) holcls and.let / be a somewhere d.ense subset continuous' of. X. Then intcl/(intcl'4) c intc$(clA) +A' Since / is (ii) is that so / fopen' Since J{clA) + clf(A}. Ifence, intcl/(4) lA, impties (i), we have that if (ii) holds, / is dopen'
Nowassumethat(iii)hold.sand'let.llbeanowheretlensesubset d"ense of Y. For any set U, open in X, int/(U) +A' Since l[.is nowhere : fr' nN in Y, there is a set T[, open in Y, such that Tf c int/( U) and W :O' But No*, iotl'(w) +6'. Since / is injective, int/i(17)n/t($) int/r(Wj  U. Ilence, /'(}[) is nowhere dense in X'
a' spa'co conor,r,A3v 4.5. LetJbe a continuous functi,on fronl a spa,ca x ittto Y. Ttten ! is 6o7ten i,f and, only if tke i'maga of each nonemgfiy open subset of X i,s somewhera d,ense 'i'n T. ' The class of all open function d"oes not contain eYely dopen function' function nor does the class oi ail dopen functions contain eve y open,numbers' real of set as the following example wi1l illustrate. I.,et X be the topology o' on x be generated ancl let g be lhausualiopology on x. Letthe is not a Baire ltyf v{AnQl aril. rqow &,r) is a Baire spacet (X'o} open' To see hbat i' space, and the iaeniity function i': (X',f)>{X'6)is tret,ween 0 ancl 1' is not dopen, let .4 be the set of irrational numbers (x, o). rlowevert of Then ciol" : _4. so t;nat, A is a nowhere d"ense subset subset ot (X'{)' intvclrA: (0r 1) 'qo t'hat' A is a sornewhere dense not feebly open' If T5us, z is not d;p;;. Cf.*"fy, r:1 is contin*ous but
Baire
sp;rees
3 is a nowhere dense subset of, (x,gl, then cruB d,oes not contain e n(*, b for any a, beX. Since cloB c. elrB, intocloB :0. Thusr B is a nowhere d.ense subset of (X, o) so that at is dopen. \Ye will now see how Baire spaces and s,paces of second. category
behave und.er dopen functions. Pnoposrrro* 4.6. rf f is a 6open function from a sgtaco of second, category X onto a space Y, then T ,is a sgtace of seconr catigor.,g. Proof. suppose that' T can be representecl as the countable union
of nowhere
clense subsets, sa.v
x
: f'(y) :
f : iifr. i:7
Wo*
f,t 0ar,t :
.*4
c! E t,'
3,
Jf'tr,t. :t
X: {(u,U)rE,l g +O}v{(n,0),Ezl n.e} with the topology inherited from 82, and ret y tre the disjoint topological sum of {(n,g), E'l u ;e 0} ancl {@, V, Ezl n,0}. rt is easy to see that x is a Baire space, Y is not a Baire space, and the identity function from.x dopen.
TrrnoRnm 4.7. rf f i,s a Jeebtg continuous 6oTtetz fnncti,on from a Bai,ro X onto a, spaca Y, then T i,s a, Ba,it.e space.
spaco
Proof. suppose tb,at v is an open first category s*bset of y. since / ft (Y) is of first category in x. since / is feebt_1' continuo*s, there is a set U, open in X, such that U.,fr(Tzj. fnus, U is an open first category subset of X. rn view of the similarities of pseudocomplete spaces and Baire spacesr one is tempted to try to characterize Baire spaces as some type of continuous images of pseudocomplete spaces. we witi now see that there is no such charactefization using continu.ous dopen functions. Pnoposrrrox 4.8. Let f: x>y and, g: B+? be continuous 6o7ten is
i;t
34
Thus, X is of first category since f ,lt, "n"". Baire spaces. tror example, rrowever, dopen functions d.o not preserve Iet
into T is
r,,I
dopen,
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 6oTtenfu,nction'. Proof. Tlet 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 intsltr'1(B). Since/ and g are dopen, 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 xtr[. 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_F1(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
clF1(clB). Ilence, I{n,s)€clB^(Xt xl[) so tb'at {y,t)ecIB. Therefore, tintcU(U)lfintctg(7)l  clB. E is dopen 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'ocomgtl'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 dopen function f: Y>x. Define n: TxY>XxXby n@,U):(f(n),f@)J for each n,!eY' Now YxY
is a pseud.ocomplete space [52], anii hence, a Baire space. By Proposition 4.8, F is a continuous dopen 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(XUl 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 otfl(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 pseudobase 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,obase 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 fr {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 dense 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.obase
* 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 1fr(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:
@
nl
M) nZ. lTow is a closed" cover for n Since ft(z). /r(e) is tE*ofL(z)l =7,2,...) .of second. category in itself, there is a positive integer k snch that
$1.a.
.
proof.
LFk^fr(z)l # O. Thus,
ze
r
,"1
f&e{F.
,.i , '.'' *!@ :
..
li,
{@
".ry #:l
Since Y is o{ second" category, there is a point ze (T
intyr1ay
.
:
IL This contrad,iction completes the
Conor.r,Ea,y 4.72. I'et f ba a conX,itr,tr,ous open fu,nction fro*r, a, spa,co X wi,th a cauntable psattd,abase onto a Bai,re space Y. If there'i,s u subset Z of Y such tkat y  Z i,s of first category i,n T and, f'(r) 'is a Baire space Jor each ze Z, then X is cr, Bai,re space.
ft is shown in [46] that the converses to Theorem 4.11 and Corollary 4.72 ate false. That is, an example is constructed (nsing the continuum hypothesis or Martin's axiorn) of a separable metrizable Baire s:pace X which has a d.ecomposition into closed" subspaces each of whieh is of first category. The natural projeotion / onto the cluotient space Y formed" b1' this d.ecomposition turns ou.t to be open map. Therefore Y is a Baire space, but /r(g) is of first category for every y in T.
#
*:
,:: '::ri
s{s ,
i
;,,;,1
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ryry Flr.:gf
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IY. The tl.ynamics of tsaire
2. Baire
*{re fi*::r i,
l
9,t$ 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: UrYa, 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 afarsorabl'a)' Tsnonna,r 4t5. 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 ct
I
Baire
exists an integel
spaces
i, such tlnat UinHAIr,, + 6. Let U, be open in X sucli
i:t t
that I/inE c (J*nH^rl/or.
ii:o.:;
by incluction, suppose there are clistinct ,ir, .. ., ,i,,_, ,ant1" ,rtt.f Ur,..., Un\ of open sets in,Ysuch that UinE  (Ui,,nA)n (avo,) j:r for er.ery 2 < /r(ri. Since {Vti'i :7,2,...} is not locally finite at any Proceeding
.:*i.l,ri''
&*
iJr :, ;!:);
clistinct form the elements of {i,r, ... Un*, be open in X such that such tha;t TJl,aHolrt, [email protected],
poin1, of H, there exists an
integel
."".;..:'
ri,,
..., i,*r] t,r) so that II,i.,^ H c {iiol/AIrt,,. Thus. Ui*,n H . (AiaH)^ !Q {I,,}1, i.q, therefore, clefirrecl by intluction. \ow fl (UinH): O since {l'; j :7,2,...} ir point finite. Also {U,) i:7,2,...} has the finite
.::'+i. .i.:r:rl,
i:. ::'ii::
;,
intelsection proper't'1 since {Ui' i, :7,2,...} c1oes. Itet fro be the open filtel on f generatecl b5 the opel filtei snbbase {Uti i :1,2,...}, ancl iet [n be any open subset of T suc]r that Fs, U';. Therefore. there is
a f iuite
.,
ii:,
ttjq,
Uo' rlence, , ,,"} such thal, lr'',, (fl LI,..)* c tf so that Ui'nX =G. I'o is, therefore, a limit of. H. Tlrns, ,0,,=](Ltin11) since E is closecl. This contradiction establishes tlN
sequence {U,,
...
LT
:i
*''
ill
;,.
:r:.::, .l
J'
riP,rlj:j
thrri. *X(F) is a Baire space in the strong sense. Conor,r,lnv 4.76. E'r;ery to,pological space X i,s a dense subspace of sone colnpact Bai,t"e sltace itt, tlte strong sense. Proof. Let Y be the onepoint compactification of X(!), where 3 ir the set, of all open filters on I. Let EI be any clo"\ec1 subset of I. Sirice X(7) is a Baire space in the stlong $ense, HnX(I) is a Baire space b1' Ploposition 2.1. There{ore, H is a Baire space by Proposition 1.19, so th:r,t f rnust be a Baire space i.r th..' stro:o ser s.. It is eas1. to see that it F is the set of all open filters or the set of all oiren nltrafilters on a "qlrace X, then ,T (3) is a gerLeral,ized, absolu,tely closecl spa.ce; that is, erler) open filter on f(F') iras an aclherent point. Tnoonnu 4.77. Erery toytologicttl sgtace X ,is ct, closed nortthere dense subset of some general'iaed absol'utely closed Bctire space. Proof. L,et rY(r,r) : i\ru{ro} be the onepoint compactification of N. Sets of the form tlx{ri} ol t'x [], r,..,] where u is open in X, n,r lf, and Itt. r,;l : {n,,n,1'1,..., (r} comprise an open base for the product topology on Z : X x'\'I(ro). I.,et ? be tlie set of all open ultrafilters on Z. B:: Theorem +.1+, Z(I) is a Baire space, Noir (fxlf)" : (XxAr)uF is an open clense snbset of Z(?). Therefole { x {o} is a closecl nowhere clense ,.:ribset of Z (P).Since the natural function frotn ,I onto X x {c'r} is a homeomorphism, ,Y is a ciosecl nowhere dense sui;set of. Z(I). Ilelrlich [30] gives an example of a llausclorffclosecl space w]rich
,nni:::.
.$4g1.#
''...rir
rrt:l.l
if.;., i;l&ai ..
I:'
r;'*+.t: t!,i.{,]:i .ii:tirlr

l**:
r:{
,ilt**1:;
at,'
IV. The ,:iaelr
end
s
ir r I I tjl
i'*ay {ec'..
,tha* :.$Aat ,'. *
.$3ece .:., ,
f.t*ite .*3:en
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.
P,Fe rs g.
aY rll  li.r *
#{ 9. :
ffi{shq$ fl rl.
.t '
mr of s{l€re
wr. i: l
'
.*.I}aCe .,ir$19,, tl:
ali '..sased ;.:,.,..
iatr*exe i.
i'
,l
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'
qqfsm '*lense
66* of +&r'tul. rii: !
.. ..
:
:wbictr
clynamics
of Baire
spaces
5l
is not a Baire space. Such an exanrple can be obtainecl by taking the unit interval and furstead. of the usual topology f , giving in the topology {v{LrnQl a,9}. This space cannot be a d.ense subspace of any Ilausd,orff Baire space. Note that X could neYel be regular Since a reguiar Ilausdorffclosed. space must be compact, and., hence, a Saire space. It is still unknown if every regular space is a dense subspace oJ some regular Baire space, though Krom l35l has demonstrated. thatevely normal regular space is a tlense subspaee of some regular Baire spa,ce. Of co6rse, el'ery T)'chonoff space is a d.ense subspace of some ilychonoff Baire space. In la?] it is shown that every regular (not necessarily ?r) space has a regular Baire space extension if anil on1;r if eYery ?, space has a ?, Baire space extension. Afso eYery completely regular (not necessarily ?') space has a completely regular Baire space extension, ancl every 7r space has a ?, Baire space extension. This paper also gives necessa y ancl sufficient conditions for a flausd"orff space to have a Ilausdorff Baire space extension. Reeei 155] gives necessary ancl sufficient cond.itions for a Moore space . to be tlensely embed"d.ed in a Moore Baire space, and" to be clensely embecld.ed. in a elevelopable llausd.orff Baire space. Aarts and" Lutzer ploYe a pseud.ocomplete extension theorem in [3] which is analogous to Theorem 4.14. \Ye give the following version of it. Tsnonnrvr 4.\8. If F is an ad,mi,si'bl'e set of open ultraJi'ltrers on the qu'as'i' fbgular space X, then X(n) 'is pseudocomplote. Conor,r,lnv 4.19. Eaerg quas'iregu,l'ar spa,ce 'is a d,ense su'bspace of some pseud,ocomPlete speco.
Clonor,r,aav 4,20. Eoerg tluasi,regul&r space i,s subset of .some pseud,ocomplete space.
a closed, nowhero
d'ense
If X is a topoiogical space, define sX : X@), where F is the smallest admisible open filters on X, or equivalently, Fr is tire set of open of set filters on X generated. by the countably infinite point finite monotone d.ecreasing open filter bases on X. By Propo'sition 4.I4, sX is always a Baire space. In fact it can be shown that' fl"sX" is a Baire space for any famill. of spaces {X"}. I'et X be a subset of the topological space Y. We say that T is fdrst coumtable outs,ide of X rt for each point r< y  X there is a countable collection 0 of open subsets of Y containing r such that eYery open sub.ret of Y containing r contains sone element' of. 0. l.et A and B be sutrsets of the topological space x. \\re say that .4 is Ttoint segtcu'ated from B if for each a.A and. be B there is an open set containing a tbat' d.oes not contain b. TnuoS,pM 4.21,. Iot" amy toltolog'ical space x tha follaai,ng are tt'ue: (i) sX i,s first cou'ntable outsid,e of X' (ii) sX X i's Io and, po'int separated' from X.
Baire
52
sPaces
F x, on base filter open point finite
is generatecl by 3 countabl)'infinite say {uol i.:1t2,...}. If tr is open there is a positive integer j such that rlence, Ue E. in X ancl F e sLi, then :L,2,..'} is a countable TJl c TJ. Therefole, fre sui c s.pso that {sunt i, x. Now for any point, of outsirle base at .F. Thus, sx is tirst countable re ,f there is a positive integer n, such lhat n$ [r. Therefoler stl.r. contains F b11t d,oes not contain in so that sX  X is point separatecl flom 'Y' Tf F and. I are clistinct elements of sxx, then n'ithout ioss of generality we can assume that there is a set u, open in x, su.ch that u is continue d. in g ancl not contained in 9. Therefo'e, .F e s(J ancl '9 I sU. Thus sXX is fo. conor,lanv 4.22. Eueryf,it'stcotttztablespacais a d,ettse subspace oJ sont'e
?roof. It Fesxx)
t]Ilen
:!:i:
J,irst countable Ba'ire spfice.
A subset u of the topological space x is regularoperz if intclu : u. x i.q sem,iregmlnr rt the collection of al1 regularopen sets is a base fol the topologr' on X. Tnnonpu +.23. If Y i,s a sem,iregulaT entonsiott' of th,e space x s'urh that (i) Y zs fit'st eountable otLtsi'de of X, (ii) y X is T, attil ptoitti separated from X, tlten, Y catz, be embeddetl as a su'bspace of sX corfia'in"tng X' proof. For each ,!J.Y X there is a countable nonotone clecreasing Y, 1rase, {T/;(E)l d :1,2,...\, at y consisting of regulalopen subsets of g(y) generatecl x on filter open be the I_,et Uo1g1 : v/g)nx, ancl let b;, the open filter base {uukill i:L,2,...t\. For each re x t'he1e is a set 7, open in x, such that sTr contains F(11) b[h d.oes not contain r since YX is point separated from X' Ilence, n1 Y andthere is a Positive
k such that ti1,('y) c V. Therefore, nd Ux(U) so that ]Ort l :0. flius, {uu@}l ,i:7,2,...} i.r countably infinite and, point finite. or each nex let f {r) : r, ancl for each Y,Tx let f(u) :gQl)' To see that / is injective,let y and s be d.istinct points of Ix' Since YX is ?0, we ma)'assume wit'hout loss of generality that there is a set TZ, open in I, ri.hich contains y but cloes not contain s. Thele is a positiye integer j such t'h^t Vi(Y) c Ir so thah zlVi@)' n 'q\l) :g(z) there woulcl be a positive integer k such that Up(z) c llj(?l)'
t::
integer
NowV1,@)c.%(y)*ot]rataeYi(y)'Thiscontradictionestablislresthat
"f is iniective.
tr be an open subset of X containing r, and 1et'['b: an geYXthentirereis apositive opensubset of Ysuch that u :vnx.If Ui(g) c T] so t'h'at' 9(y)<sU' c'[,'. Ilence, j YiQ/) iriteger snch tt.at ancl [/ be open in X such t']rat yeyX \et Therefor:e, f Jr) c. sfi. Now n .F@)rsU. Then there is a positive integer such that U,,(y) c [r' If
llt
ne
X,let
.# g
i; :;1
;il
IY. The clYnamics of Baire
t V"\U)' zeVr,{g)X, then there is a positive integer k such tinat' Vo@) c sU so that
ffi*ite
fherefore,
.PPen t''Sbat wxltrie
U1,@)
c. U,(3/) so ti$g(aJesU' Thus,
/
flv"(y)l
is continuous.
Toseethat/:Tf(Y)isopenlet'VbeabasicopensetinY'ancl
5eint x**ins .t1
SSs
o{
s€.6
tr
{c*r. ril,
tl,,,
f.ryre ..'.
T1
W t*.e ,l
ot
spaces
is a posiIet U :VoX.Tf n
3.
,t*nl
Hyperspaces
anil functicn
spaces
AthoroughinvestigationofwhathappenstoBairespacesinthe
g
id
T
*ffit€d
iws {6*ia
is
*
ffS*u
$.r'lrt
,ryt" s3
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*:'*&ere
liffFere ,
F{v)
i.,4qEi. Ni t'bftt
present formation of hyperspr""I* "url be found in [a1]' \{e wili now is a topological spacet some o{ tne resutts obtainecl in this paper' It (X ,f) X ancl 6(X) denot'e of subsets Iet 2x d"enote the set of all nonempty closett u1,"',An'{,t'hen 1.fthe set of atl nonempty "o*pr"i subsets of X' and AaAo t'A for each i' : A(ar,..., Un) :
{Aezxl
2O,
topThe finite topology (or Vietoris topology) on 2x is the' :.L, "'' rz}'fhe ology generated by th; ba;; {(Ur, ..., Un)l aaef ; 'i base {(U1' "'' An)n topology on 6(X) is the topology generatect by the UaeV, i' : 7, ..', n\' ^g(X)l Tsnosnu 4'24. If G(X) i,s a Ba'i're space (sytace of second' categorg' resp')' resgt.), then X i,s a Bu'ire sytuce (sYtaca of socond' category, o'^ T onnm 4.25. If X i,s a Irsgtace and' 2x i's a Bui're space (space categorEf second' of (sytace seeo'pd, eatregory, ,rr1,.i, then X i i nn;'re sgtace
1,...,n).
rasp.).
Proof.IfXisnotaBairespacethenthereisasetU,openinXand Uul : A' a sequence {U} ot dense open subsets of X such that O^fl T'et X 1 e V nU n of . . ., Y n)be a bt'sic open subset T/Jt k el[ and let ( ?"'2x' It can easily is a ?rspaZe, {nr, "', %}' ''1, for each j : 1, ...,il.Since'X that so ;; ;;;; tbat {n'i...,uo} is cont'ainedin (u*)n(Y""''vn) (U*) is dense in 2x. Clearly, (U)n (h<arr:A so that' 2x could" n
$u,€rt an
g*xi*he $lp'" *U. p,& tJrat F..
f. rf
not bo a Baire
spaee.
5{
Baire splc,rs
topology {" ott the s.t X' is definecl in [41] which lies strictly between the Tychonoff piocluct topologS' ancl t'he box topology. Using method.s similar to those used in proving Theorem 3.16? it is shown that if X is quasiregular ancl (X' ,7u) is a Baile space, then 9x i.q a Baire space. It is also shoryn that if .I is quasiregular and. C(X) is a Baire space, then (X',{"} is a Baire spa,ce. ft is easy to see that for a ?rspace X, if either 2x or C{X) i,s a Baire sllace in the strong sen\e, then so is X. ft can also be shorvn that if X is pseuclocornplete, then so is Jr. Function spaceq are not so well behaved. rvith respect to the property of being a Baire space. ff X and Y are topological spaces, the set of all continnous {unctions frorn X into Y rvill be d.enoted by C(X, Y). In tlre case that (Y, d) is a bounclecl metric, Cu(X, Y) witl represent' tire rnetric space (C(X,y),A), where a it th* supremun metric on C(X, Y) inclucecl bv d. Also C(X, Y) uncler the compactopen topolog.v (resp. the topology of pointwise conlrerg'ence) will be clenoted by Cn(,ll, Y) (resp. Cu(X, Y)). fake I to be the closed interval from 0 to 1 n'itir the nsual topology, Iet I() be the rationals in f, ancl let In be the irrationals in f. Define Z to be the set (frxl)u(foxlB) rvith the topoiogJ'inheritecl from the usual topolog; on f x1. Now Z is a pseudocomplete, separable nretric space, but Cs(I, Z), CkG, Z), and Cp{I1Z} are all of first categor;' (see [43] ancl l44l). On the other hancl,if f," : Z witlt the topology generatecl from the same topolog;' xo6 the addition of 1nx16 as an open subset, t]nen Z* is of first categorJ. while Cu(f ,Z*) and. C,(I,Z") are of second category. But Co(Ir Zu1 is of first category. Thus to get a similar example for the topology of pointwise convergence, let I"be I with the countable cornplement topology and. let Zi be (InxI")u(IBxlo) with the topology inherited" from 1 x f" and" the ad.dition of /n'x fo as an open set. Then Zi is of. first categorS', while Cn(I", Zll can be shown to be of seconcl categorS.. tr'or an arbitrar.v interval 1, cond"itions which guarantee tbat CrQ, Y) is pseud.ocornplete may be found in [a5]. lVe also find that for any locall5' pathwise connected. metric space Y, if" CkQ, Y) is pseudocomplete, then T is pseud.ocomplete. Sufficient conclitions for Cn(X, Y) to be of first categorS' can be found in [aa]. \Ye state two such conclitionq antl indicate the proof of the sirnplest. A topological space X ls saicl to be completely Eau,sclorff tai,th, respect to Y provicled. that for eyery finite set {nr, . . ., nn) of distinct points of X and every finite set {Vr,...,YnI of nonempty open subsets of Y, thele exists a continuous function f : X>Y such that f(n1)
A
i,:1t...t11. 'Irmonnu 1.26. If X is completely Hausd,orJf with respect to Y Cn(X, Y7 ,is a Bai,re space, th,en Y is a Baire spaae.
and,
7:'
IV. The ilynamics of Baire
Proof.
**tg
spaoes
X is completely llausd"orff with respect to Y, Co(X, y) subspace ot,An*, where I, is a copy of Y. Therefore, by Since
x.irg
is a
ttua*
Th:orem 7.15, Il Y, is a Baire space soihat Y is a Baire space.
tre. &e.n FErer
,si*s
*s*r r*1r I
In
,&he
'F) *rlt, *he
,
*e*Ls i}{e,{L
eb,le
pry H€"11
Fglea t::
*0r $A&r ;'
'*k€ #it& .:..
.
ryn 'e
*f
ltT'\ :StIT *&e:r
ies$ ;:
tur. fu F
€x *aere
r,e3I:
.
g,!*t*
dense
oo
f
e
,T
Tsnonnu 4.27. If X is a n'ond"isu'ete Trspace whi,ch' i,s completely Hausd"orff wi,tlt, respect ta Hau;dorff space $,.then Ce(X, Y) as of f,irst i : . r i '.:> 'z',',.;.3 : ..,. .,... ca,tegorA, Sufficient conclition f.or Cr(X, Y) to be of first categor'1' can be founcl in [a3]. \{e state two sP:cial cases. TrTnoRpM 4.28. Let ilI be an nmanifold', ancl Let Y be an (n7)contzected,,localtg (tr,7)connected' metric spa,ce. Th'en' i'f T i,s of fi,rst categorg, so i,s Cp(M, Y). TrrsonsM 4.29. Let X be a metric,sl,&ce, and' let E be al'ocal'l,y conues: Li,near toytol,ogi,cal, sp&ce. Tluen i'f E i,s of fi'r:st catego?'g' so i's C,,(X, E).
V. Products of Baire spaces One
of the most clitficult
questions concerning Baire spaces has
been the question as to when the prod.uct of Baire spaces is a Baire space. In this chapter we iclentify several spaces whose product with a Baire space is a Ba,ire space. It is also shown that the prod.uct of any family of Baire spaces, each of which has a countable pseuclobase is a Baire space.
!
The notion of a kBaire space [63] is cliscussed. and sufficient conditions are given for a space to contain a d.ense Baire subspace whose square is of filst categorI' in itself. \Ye then sketch Oxtoliy's example of a completely regrrlar Baire space .whose square is not a Baire space. I'inally, we show how Krom nses Oxtoby's example to construct a metriza,ble Baire ijpace whose square is not a Baile space.
.
l.
Finite products It is ciear that a product space can be a Baire space only if each of its factors is a Baire space. ft is the converse of this statement that has presentecl so much difficulty. As an immecliate consequence of Corollary 4.72 we have that the product of two separable metric Baire spaces is a Baire space. Aarts and. Lutzer [2] construct a separable, pseud.ocomplete, rnetrizable Baire space in the strong sense whose product with itself is not a Baire space in the strong sense. I.iatel on we will see that there does exist a metrizable Baire spa,ce whose square is not a Baire space, Therefore, we are concerned. with fincling the weahest possible conclitions which preserve Baire space$ under prod.ucts. One clirection is to ask which properties are such that the prod.uct of a space with one of these properties and. a Baire space is a Baire space.
Tnuonnivr 5.I. Let X and, Y be Bai,re sp&ces. Then X i,f er?,y ane of the fol,lomi,ng cowd,itions is sal,,isfiedz
spa,ee
(i) (ii) (iii) (iv)
X as fi,ni,te. X i,s toealtl,lS afaaorable. X,is.
X
pseu,d,ocom,plete.
'is gu,asi,regular ond, Eausd,orffcl,osed,.
xT
i,s
a Baire
!
Y, Products of Baire
sp&ces
(v) x 'is guasi,regular and, Urgshonelosed,. (vi) X i,s (regular Hausd,orff)closed,. (vii) X has u, lacull,y countablo pseuclobase. (viii) X ,is second, countabl,e. Proof. Ii X is finite then XxT: U({r}xy) is a Baire
#w
iras
::. l
.:
P:8pace.
4 tsaire i,f,*mily * Baire ,'
wx6
*on
nrsqnaTe
is
eour
s;Ilauy, ::kieable
;$f *ach @*
tlat
9,.€*"tS,,$Fa{es
pe.xclo$Eeduct l
*_
uFlLu
d s'Itn uader h, procl&
m,Sa.ire 'r,
*,l.8*",
space
r<x by Proposition 1.19. The proof ot (ii) is due to White l?01. Let {/u} be the sequence of functions guaranteed by the clefinition of a weakiy ofavorable space, and let {Cr} b;" a sequence of dense open subsets of X x Y. ft will suffice
to show that f^) h * 6 since both the concept of Baire space and. weakly afavorable are hered.itarily open properties. By Zorn,s lemma there is a maximal pairwise d"isjoint coliect'ion af , of nonempty open subsets of X such that for each set He &f l there is a nonemptS'set yr(_E), open in Y, with Hxyt(H) c G1. fi \)tr, is not d.ense in X, then there wouldexist a set U, open in .X, such that Un(U tf r) : fr. L,et V xW be a nonempty basic open subset of XxY contained. in er;l(U)nflr. ff we let yJV) :'tr[, then V ,yr(W)  G, which contradicts the maximality of #r. Thus, Utrr is dense in Z. I,et ff o : {)t}. Proceecling by inductiorl, suppose that the finite sequences {*tl i, 1,... k} ancl {yol i, :I,...,1t} have been so d,efined that the following are true for each ,rb:L,...rk: (a) &" is a pairwise clisjoint collection of nonempty open subsets oI X; (b) U//" is clense in X; (c) K" refines ffn_t) (d) For eacln Ee lfnthere is a nonempty set %(E), open in y, with H xy"(E)  G,i (e) If Eietri for j : I,...,n and. Ej+r E, for j :I,...,tuI, tlrcn (yr(Hr), . .., y*(E,D is in the domain of fn. By Zornts lemma there exists a maximal collection ffon, of nonempty open subsets of X such that properties (a), (c), (d), and (e) iisted above holcl for n:k+1. rf l)tro*, is not dense in x, then there woulcl exist a, set U, open in X, such thah U n(Utro*r) : O. tr'or eacb 'i : 1, ..., k, there exists a set E[6eKi such that g + UnEkc. Enc. Hrr_t...  Et. T'et V xW be a nonempty basic open subset of X xT contained. in I(u^Hilxfo(y,(E,),..., vn(Hn))lne**,. rt we let Tn+r(v)  w, then V xy**r(V)  G,,*, and (yr(E ), ..., T*(Et), yt *r(V)l is in the domain of Jx+, which contrad.icts the maximality of #**r. By the principle of finite induction we have sequences {af,nl ne N} and {ynl ne N} so that for each rae lf properties (a) throug'h (e) listed above are true. {o* n ll)x"l tt,u Nl # 6 sinee X is a Baire space. If u. fl lU*" tt,u Nl, then there is a sequence {Eu} such that for each ,i, neEu and. Ea*, ;1
58
Raire spaccs
c Ht.Thelefore, foi: each ,i, (yr(Hr),...,Ti(H)) is in tire dorirain of /n so tlrat Oy,(Ho) +6. It ye nyi(H), then 7:l
Xl
(n,
a).4 tt, x h(H)l i:f; G,. i:7 r
Since e'rery pseudocornplete space is neakly afavolable, (iii) is immecliate. Norv sullllose that 'f is cluasiregular T{ausclorffc1osec1, ancl let F be the set of all nonconrergent open ultrafilters on X. Since .T(?) is Ilausd.orff 1331, X : X(I). Theorern 4.18 shows us that X(F) is pserrd.ocomplete.
For each'i,Iet 0; be the topolog_v on X.If {Ur} is a seclrience of sets rvith Uoe 9o and. cl(ii*, c Uo for each ,1, then {tlo} is a regular filter base on X. Ilerrlich l30l has shown that, in a Urygshnclosecl space, ever\ur.vsohn filter base has an aclherent point. Banaschewski l7l has shown that, in a (regular Ilausclorff)closecl space, e\rer.l regular filter base has an ad.herent point. rn either case, [) ui *a so that x is pseuclooo1r]
i:r
plete.
Part (r'ii) is an immecliate consequence of Theoren 1.11. Clea,rlv, everJi seconcl countable space has a localll' countable pseud.obase. For a procluct space to be a Baire space, it is sometirnes necessall. to initiall"v lequire the product space to possess certain properties. 1l.e will now introcluce a concept whic]r falls into this categor_l'. A space ,r will be callecl a generati,rtg Bai,re space providecl that whenever ql is a countalrle farnily of open subsets of x, then there exists a countable familj. { of open subsets of E such that 4/ u{ generates a Baire space topology on {. Pnoposnrou 5.2. Et:ery generat,irtg Baire
s,pace ,is
a Ba,ire space.
Proof. Suppose that (X,,7) is not a Baire space. ?hen there exists a ue{ such that u: 0/r, where eac)t anis a closed nowhere d.ense subset of X. Let ql : i;'Anl nunflu{U}. Now let 4" tre an arbitrary each.
A,, is closed
, .Tt (X, r) Baile
,f,
and.let z be the topologv on X generatecl by Qtv{. Since in (X, z) ancl nowhere d.ense in (X,t), and, .qince then each An is nowhere clense in (X, z). Thus since Uet, is not a Baire space. Therefore (X ,.q) is not a generating
subset of
space.
i
.ir.
1r
Trrponprr 5.3. Let X be a Bo,i,re space wnd, tet T be a gen,erat'ing Ba,ire spa,ce. If XxY h,as tlte cou,ntable ch,ain, cond,ition,, then XxY zs a Bui,ro space.
i
59
V. Procluots of Bairc spaccs
Baire
Ploof. Suppose that X x Y with prod'uct topology{ were not a where O: space, so that there would' exist a U ef such that ]rn*' of XxY. hen for each n,
,tid ;nfl ll
jF] Ss.., @*,1 g+.
*:::,s lr' "
ryrl &as
iffiIrlr
*: .,.:..
i,.
,,F TF
a
s$lx
*t
n" is the projectionmapofX'xYontoY.SinceYisageneratingBaireSpace' of Y such that qt is contained. in U A,,.1{ow 1et : {nv(Ul")l n,r,e ff},where
there exists a countable set l. of nonempty open subsets z be the product 4/v'{ generates a Baire space topology 7Y oL T' TLet' Baire sp*3e since (Y'rv) topology on Xx(f, zv;. iVow (ixY, z) is a since each has a countable trase. For each neN, el"Anc XXnl)rUi
tld,rr. Since 1] Ui is
dense
in XxTAn with respect to {
and'
A"
dense in (X XT r)' is nowhere dense in (X x T tg)l then cl"l, is nowhere ' (XtsY' 1) ry"q z, then of fhus since U contains a nonempty "te*bet (x xy rtr) not be a Baire space. llhis contradiction then estalolishes that
a Baire space. Itisnotd'ifticuitto'seethateveryopensubspaceofagenerating space Baire spa,cer is a generating Baire spacet and also every Baire latter The space' Baire haing a countabtJ pseuAoUase is a generating the topoiogy follows from the tact ihat tf I is a pseud.obase for X, then if X is a B*yu on X generated. by I is aBaire space topology if anil only Spaceunclertheorigina}topology(see[4?]).Becauseoftheexamples to come in section 1, ***o*iogl th" continuum hypothesis or Martin's axiom,thereexistsaBairespacehavingthecountablechaincong"o*r^ting Baire lspace. Ilowever, the followd_ition which is not ^ ing is true.
is
ind.eed
TT{FoR,EMs.4.Euergalmostcountabl,ycornltl,etesTlacehaa,i,ngth,ecountable :fux.ts
lry
;a &*Y .,rt""'
ffe€ @#
'*o,
ffieg
,ry
wqs
chain cond,iti,on 'is a generati,ng Bui're space' 'Proof. tLet {Xrtr) be an almost countably complete space having
CCCancL?et{9,}beasintheaboved'efinition'IrctQ/:{A*ln'el{}be without a countable family of open subsets of X, which we may assume CCC' has X Since intersections' finite loss of generality io n" Jto.*"cl uncLer foreachrzthereexistsacountablefamilyg!cZ'suchthattheclosure is d"ense in U"' Set of eaclr member @
Or: U go*r";; sections
" gT.Now
*f
is contained"
in
U,, and"
l)9i
0r, "',4n have been d'efined"; then define interfollows. T,et' {W*l ne N\ be the family of all finite gountable ot [nr. since x has ccc, for each n f;,.ere exists a suppose
58
Baire
spaces

Hr..Tlrerefore, for each i,, (yr(Er),...,Tt(E)) is
so
that )yo(Hn) +9. i: I
rc. A€
(rc,
y),
nyi(H),
i:l
Dr"rx
ln the dornairr of fi
ilren
y/E)l .in,.
since ever5. pseud"ocomplete space is rveakly afavorable, (iii) is immediate. Norv suppose that x is quasiregular r[ausdorffcloset1, ancl let Jfl be the set of all nonconvergent open nltrafilters on x. since x(?) is rfausdorff t33], x : x(r). Theorern 4.1g shows us that z(_E)is pr"oaocomplete. n'or each i,,Let'
gi be the topology on,x. Tt is a sequence of sets "with Uoe go and" clUi*, c. Uifor each i,, then {uc} {Ur} is a regular filter base on x. rlerrlich l30l has shown that, in a urysohnclosed, space, eyery urysohn filter base has an aclherent point. Banaschewski [?] has shown that, in a (regular Ilausd_orff)closecl space, errer)r regular filter base has an ad.herent point.
plete.
fn
either' .or",
ff
i1
Ui #@ so tirat X is
pseuclocorn
Part (vii) is an immecliate sonsequence of flreorem 1.11. clearly, €yery second countabie space has a locally countable pseud.obase. tr'or a product space to be a Baire space, it is sornetimes necessarl, to initially require the prod.uct space to possess certain properties. 14re will now introd.uce a concept which falls into this category. A space x will be salled a generat'ing Ba,ire sytace provid.ed. that rvhenever ql is a countable family of open subsets of x, then there exists a countabie familJ, l/ of open subsets of xsuch that elvt generates a Bair.e space topology on lf. Pnoposrrrorv 5.2. Euery generati,ng Ba,ire space is a Ba,ire space. Proof. Suppose tbab (X,t) is not a Baire space. Then there exists
e{
that A : 3 A*, wbere eaclt Anis a closed nowhere d.ense n:1 subset of X. Let oU {XAnl nrlf}u{U}. Now let t be an arbitrary subset of f, and let z hre the topology on X generated. by a//vf. Since eaclt An is closed in (X, r) and. nowhere d.ense in (X,{), and iiince t 9t then each An is nowhere d.ense in (X, z). Thus since Urz, {X, z) is not a Baile space. Therefore (X,{) is not a generating a,
U
such
Baire space.
T*ng*sivr 5.3. Let s?&ce. .
; :i
.8pa,ce.
If XxY
X
be a Bai,re space and, tet
y
ba a generating Bai,re
has the counlable clmin cond,,iti,on, then
Xxy
zs a Baire
:P
..:w i
,t'*& r' '
#
]i
' ..i::,
,.&
trr. Procluots
proof.
#f. : .r?
suppose
that
x
of Baire
spaces
59
x Y with ploduct, topology.z were not a Baire
that there would exist a U e{ such that O : PrO", where N' eacl\ Altis a closecl nowhere dense sutrset of XxY' Then for each ne
space, so
there exist a countable number of basic open subset's {uil tt€ lr} of x xY @ such that 3 U; l* a dense subset of x x T  an and. at least one ui :
ii
in
,l:
It
*.eed
i
iir;
w&*. :.a
ii
@t€ ,:
. rI at
FfiSt 'i '::
*eq5
t:
s. ]L !,i..a' '
ffi.{c$ir$}s _
r'l'
@sr a.
@t"s I
rye .l; ,
.Tj,"
g"
ffiEg
:. f#$fe *e$rs i:,.'
projectionmapofXxYontoY.SinceYisageneratingBairespace' ihere exists a countable set l' of nonempttr open subsets of Y such that Q/v{genelatesaBairespacetopology.CyoTlY.Letzbotheprod'uct (Y' zt) topology on Xx(Y, z"). Now (XxY, z) is a Baire spa:e since
is
 l
l*trc
A,,. Now let % : {o"(t}i11 tt', i'' N)1 where zv is the
.i1
,&&
$ay WF

r)' is nowhere dense in (X x y ,{)) then cI"' ,, is nowhere dense in (X xY ' would' z) (X XT , Thns since I/ contains a nonempty member of z, then (x xy,tr) that establishes then contradiction This not be a Baire space.
*pglr ,'kas
:,t
U
since each has a countable base. tr'or eaeh' meN, cT"Anc Xxy!i.Ui a:\ @ tli, r. Since [*J Ui is dense in X x Y  An with respect to { and' An
"s*r$
::,
i:r is contained. in
'
i:"'"'
ind.eed"
a Baire space.
open subspace of a generating ,Baire space, is a generating Baire space, and also eYery Baire space haYing a countabie pseud.obase is a generating Baire space. The latter follows from the fact t'}rat tt I is a pseudobase for X, then the topology on x generated by 4 is a Baire space topology if and only if z is a Baire space und.er the original topology (see [a7]). Because of the examples to come in section 4, assuming the continuum hypothesis or Martints axiom, there exists a Baire space having the countable chain condition which is not a generating Baire lspace. Ilowever, the following is true. Tnnonnu 5.4. EaarE almost countubly comli,ete spaca hats'ing the countabtre chai,n cond'ition 'is a generati,ng Bai're spaca' space having Proof. ILet (x,{) tre an almost countably complete ql : {Unl n'< N} be CCC and Iet {9*} be as in the above definition' I,et assume without may we Z, which of a countable family of open subsets x has ccc, since intersections' loss of generality to be ciosed und.er finite
It is not cl,ifficult to see that every
foreach'r,thereexistsacountablefamily/fcg,suchthattheclosure of each member of g? is cont'ained in U,, and' l)gi is d"enso in U"' Set
frr,.'.r 0, have been tlefined'; then d"efine 41,*1'ai follows. I.,et {W*l net[] be the family of all finite intergountable sections ot gA,. Since X has CCC, for eash n t'here exists a
4r: V gf .Now il
1
i:7
suppose
Baire
60
spaces
faririly 7T+t 9pt1snch that the closureof each nren'lber of gt+r is contained
{gxl
in f[,,
ke
and l)gT,+,
is
dense
in
TIr,,. Define
Or+r:
iY} thus clefinecl by incluction, then take 4r
:t1e:
,l)rg']+r.1tr'it]r
: () U*, which k:7 by Ql vf
is corintable, ancl 1et z be the topology on X generatecl . fn orcler to establish that, (X, z) is a Baire space, let {1r,rl zr,e lI} be clense open subsets of (.X, z) ancl let' V be an arbitrary nonempty open subset of z. Since 7, is ciense, Vnlr, contains a basic open subset of z. fn fact, we ca,n fincl an tmle N and B, e %ru, sa fhat c7, Bt c V n}.r. Now BraV, +9, so there exists't/L2e rY with nt,z> rrLt and Bre 8nz, sttch that tT, Brc. BrnV2.In tiris mannel rre malr intrictively define a sequence {Br} srLch that folnrt < rnz < ..., Bie g!*i ancl c7, Br*, c. BonVi*r.
Then since .F
(X,{) is
)Bo * 0, so that (X, z) is a Baire space, so tliat (X,7) is
almost countably complete,
V.,) * O. Therefore ""fl a generating Baire space.
2. Infinite
tl
products
It is weli known that the countable
product of complete metric is a cornplete metric space. Oxtoby [52] has shown that the arbitrary procluct of pseduocomplete spaces is pseud.ocomplete. Therefore, the arbitrar;' prod.rict of complete metric spaces is a Baire space even though it may fail to be a complete metric space. l'rolik [20] illustrates that the arbitrary plod"uct of cornplete metric spaces neecl not, be a Baire space in the strong sense. Ilaving a countable pseuclobase is another property such that when the factors in an arbitrary protluct ale assumecl to have this property in adclition to being Baire ,space, then the prod.uct is a Baire space 1521. To see this we need the followinE lernrnas. ' LnlrmE 5.5. Let X and, Y bo topol,ogi,cal, spaces wi,th T h,aaing u countablo pseudobase. If {G}'is a sequonce of clense open subsets i,n XxY, th,en, tho set of all poi.nts fre X, su,ch, that for some neN, (Gn)",is not a clense open su,bset of Y, i,s of fi,rst categot.g 'in X. Proof. Let {Pol ?€,lr} be a countable pseuclobase for Y. For each'd, j<l/, let E:ntl(XxPr)^Gjl which is a clense open su]:set of x' ff f A":6, t'.lnen T is of first category and we are throug'h. For any spaces
i'
!,
'.\
.:_
*{
point r< OE, it is eas;'to see that, each (Gr). intersects eachPo. Theren,j fore, (Gi), is a d.ense open subset of Y for each j. Thus, the set of all points fr, X, such that for some ne N, (G,), is not a dense open subset
4j&t;
.{.,'::
:r;& j' .
,.'
:;
V. Procluate of Baire
of y, is
;fiOtrl
r.
category in
:
lYith
:SF€I}
5.1.
z.
''l{ow
X : fl *r. For simplicity, i:L tation for 0(zra(ra:
s&ace '
For each positive integeli,rlet xobe a space with a countable pse'.rdo
base and let
:,that '!, :l.l
'
F.,,.
1L
rf
X(n\:ltx,, TT
*ka.t' ..,'.
9
;:,,
l$.'Y,*a
&*!re
l:
.t .'
t.
&"'&el
t
.@ *&e
8$&se* ,,t l,
Meh tl:
g$
eny &*xe
will
use the following no
1I
.I_Iv
A(m.n): '
ttl,
i:m*l
'i.
{rr} x Y(nr) c. UrxY(n') c. UnGr.
j
proceed.ing by incLuction, suppose that the finite sequences {nil :L,...,h\ ancl {enrl j :7,...,k} have been so clefined that the fol
j : 1, ...,k: (i) 0:no{%t<...
lowing are true for
(ii) *1e X(n1r,n1). (iii) {(rt, .. ., ni)} xT (ni) c. U n4i. (iv) n'or each n, (G)<*r,...,,i> i* a dense open subset' oI I(n). There is an integev nk+r ) ru* such that (G*nt) (q....,r1) contains a set of the form Ua;1 XY('nr*r), where U,,a1 is a nonempty ope!. subset of X{nr, nrr+t). Since X(ra7, , nn+t) is a Baire spacer U1,.1 is of second category' Appiying Iremma 5.5 to X(n*, nr,+r) xT(nn+), we get a, point fi7,1€ U1;'1 such that, for each n, (Go)<*r,...,r/,+1) is a dense open subset of Y(n'6*1)' Now
.
#
al]
eF*et tr:).
a;.'
For each ,n,, N, X(,?) has a countable pseudobase {Pf I i.lf}. ft is easy to see t]nat, {PixY(n}l n,iril} is a countable pseudobase for,Y. l{ow let {G"} be a monotone d.ecreasing sequence of d.ense open subsets of X, and. let U be a nonempty open subset of X. There exists a positive integer rq, such that UnGr contains the basic open set UtxY(nrJ, where C1 is a nonempty open subset of. X(nt). Since X(rzr) is a Baire spacer U1 is of second category. Applyrng I,emma 5.5 to X(ntlxT(nr), we get a point nreA, such that for each n, (Gn)r, is a d.ense open subset of Y(nt). Now
fir01S*re,
W*f*
Y
rn'e
(n) : il ",, i:n*l
rs
te.fric
I

;l
w*'v fssl.
wtrich is of first
u(xEi) i'i
d'i
Proof. For a finite famity of spaces it is easy to see that the plod.uct has a countable pseud.obase a,nd is therefore a Baire space by Theorem
be
.:.{3E
X.
in the set xnvi:
Ijslv(}rA 5.6. Ttr,e Tgchonoff prod,uct of ang countable fami'ly of spaces, eaala of whi,ch, h,as a countable pseud,obase, has a countable pseu,dobase' Iurthormora, 'if euch space 'i,s a Bn'ire sp0,c0, th,en the prod'uct'is a Ba'ire sp&ce.
rbich .hr$
contained.
61
spaces
{r**r} xY(nr*r) c Ur+txY(n'r*t) c
(G7,11)qr1,...,o7r).
Baire
{i2
spaces
Theleforc.
{(nr, ...,
nr,+,.)}
xI
(nr+t)
c Gr_rnG.
B;' the principle of finite induction we have the sequences {nr l j. ]I} ancl {rtl j.lf} so that for each jeil, 0 :1xo
':,'
"llX" Proof. Let {G"} be a sequence of dense open subsets of X : IIZ". By Zorn's lemma, each Gn contains a maximal pairwise disjoint ?:*Ltty of basic open sets, {A},1 me Ar}, which is countatrle since X has the counta
;.
'..i.
a .!:.
ble chain condition. Therefore,
j. i:l
it,
ii
?1
ir,
snb,.et of fi_
:I:
,17L
a<
*
$.,
.,*:1,
out
o'B
in
fiX"
Thus,
{$
ancl, there ore, fl G,,, is clense
'
.if
$i $,.
ji
i .ll
ii
t
i:
clense
since
flX"
is a Baire space. Ilence, OH,,t
in X.
A topological space X is said to have cali,ber f{, if each uncountable faniil;' of nonempty open subsets of X contains an uncoultable subfamily with a nonempty intersection. rt is easy to see every separable space has caliber ltr. and that ever;.
$t
.t
DrK"is
nt
,*
t'.i: jtl
At B
X". Since each 8, is dense in X, each K, is dense in f]X".
S,l
;ii
o.
Aftbe this tinite suiset of 4, and ]et ,B repre,sent the countable set l)A';,. Now each .E" is of the form iK,,x Il X" where Jf,, is an open
:i,
t;
all but finitely many
TLet
il,
,
a clense open suirset of X.
"":,PrUftis Eacb Ufr is of the form 1l Uo, rvhere U o : Xofor
nl
space having caliber ,rt. hasr the countable chain condition.
I,nulrq. 5.8. The product
.
oJ ang
faruily of spaces,
luhle pseud,obase, has the cou,ntable cha'in
each of wlti,chhas a coLtrl,
cond,,it'ion,.
Proof. Let {X"l aeA} be an arbitrary collection of
spaces, each of which has a countable pseuelobase. clearly, any space that has a countable pseudobase is separable. Therefore, each X o lns caliber t{, . Sanin l58l provecl the concept of having caliber N, is prod.uctirre. Thus l/X" has caliber l{r, and. hence has the countatrle chain condition. aea Tsroa,nu 5.9. The prod,uct of ang fami,ly of Bai,re specesj each of whi,ah ltas a cou,ntabla pseud,obase, ,is a Ba'ire sp&ce. The next proposition also follows from the argrrments used, in the prececling lemmas.
#
ril
:t
#
{i:ii
ff' f:'. 3,
$,
\i.
Products of Baire spaces
Fr $
lf]
,$"
sd {ii)
set d
ld,ile
$r" .ll
ffg". t'{
Pnoposrtron 5.10. If X i,s a Ba'ire space wi,th a cottntnbl'e pseud'obaso, then for any card,'inal num,ber m and for anE point ne X, tke set of all,ltoi,nts (no) e X"' s,u,ch that fio : fr for all' but cou,ntably many q. < nx 'is a Buire space.
The notion of having a countable pseud.oba,se has playecl an important role in this chapter. Clearly, eYerli secontl countable space has a countable pseuclobase and. every space with a countable pseudobase is separable.
In general
these implicat'ions cannot be reversetl, although these notions are ecluivalent in a metric space. The next proposition gives a large class of spaces which d"o not have a countable p'*eudobase. PnopoSnrON 5.11. Euerg uncountable prod'uct of nani,nd,i,scr"ete spaces, each lttr,tt,ing a countable ytsau,dobaso,
,f.
d*f
M5
ct.
f,is set ,,,,s*str ,,:,

flIr". ;fJ tF.
...s.{"' F:
!'r,::
not haae a opai'ttt cottntable psetttTo
base.
kanity
gqta
d,oes
P1o of . For each c e 4, rvhere .4 is an nncountable set, 1et
ind,iscrete space having
a countable
pseudobase, ancl
X" he a non
let
:
"
il"".
Without loss of generalitlr, suppose that the cardinality of 4 is t{t. Since each 'll" is separable, X is separable. Ilence, Xhas caliber l{1. ff X hacl a point countable pseud,ohase, therr it xould. be courtable. But it is easy to see that the uncountable product of nonincliscrete spaces cannot hare n countable pseuclobase. Trtnonorvr 5.72. If the prod'ttctof generat'ittg Bai're spaces h,as the countable cha'in cond'it'ion, then''it'is a gen,erati'ng Ba'ire sp&ce. Th'us tho prod,uct of g1enerating Ba'ire sp(Lces haaing cali'ber rr*1 ds A generati'ng Baire sps'ce. Proof. I'et {Xrl y, f} be a family of generating Baire spaces and"
: [lX, with the product topology { b.aving the countable chain f condition. L'et, Qt : {U,"i n.< 47} be a countable coliection of open subsets
let tr
r
$m,hle
@l:lF
YC
:f,
*.*rr
rvhich we rnay assume without loss of generality to be closed uncler' finite intersections. Since X has the countable chain condition, for each rz there exists a countable familS. of basic open subsets { Uf, I z, e lf,} of .T
,.wa*ri.,,
snc,h
,,
of
@
i.:t,,.,
t,:€S€.h
that prO'" is a d.ense sub.cet of Un. Now for
5" sq;€
ff#tflt. t,
s
the
,.,
:jil
tl,.
ii.:
;,1:r,
,
'i, Ui is of
tlre foi'tu I(rr,1:)
@{ti*e*r;*
each n. and'
P,
n')a''a(r/ti{"'il)
wlrere fty;(*,i) is the projection rnap from open in Xrrpr,t).Let
T' : {yi(n,'i)l
tt , t,e
,F
ancl
'
X
onto Xvi(r;il ancl
je .lr with
For erely y u l'r let
Ztr: {nr(Ui,}l
tr,,,le M}.
1
<
j < h(tt. i')}.
T/rr1rr,4
is
g
Baire
64
Since for each y, l', X, is a generating Baire space, there exists a countable set l/, of nonempty open subsets of Xrsuch that'%rv{', generates aBaire
f
spacetopology ry on Xr. tr'or each ye T',let {r: {nr'(V)l VeQ/rv'/'r}. L'et {' : \) {r, wliich is a countable set of nonempty open subsets of I,
:i
ye
l'
andlet z be the topology on 'X generated" by 4/v"f . In ord.er to establish tlnat (X, z) is a Baire space, consid.er the prod.uct topology, call it o, on X consiclered as the product Il(X,rr); where is the ind.iscrete topology on Xr. Note that 6 c r. for each ye f T', ", Since each 1Xr, %) is a second countable Baire space, t}len (X, o) is a Baire space. Therefore, it suffices to show t'hat' o is a pseud.obase for (f,, z). T'et 1Y be a basic open set in (X, z), which will be of the form T{z : Unnnit (Zt)n ... nn;)(V*), where Ane Ql and each vi€ qlyiu{y.
I
c.
q.. Int' r.
$ ,
spil,ces
'H :l:
so that V each
:T: :':
,ffi'
:
c lV. But
tli:
l;{n,il
,l
(Lf q) nn,lw,)n... n n;)tv*), since
ni1l.o(Vry*,q)
and"
*:fi
'*'. ,&,
$ $
r fl
%
yi1n,t1c r t,i(*,.ti
t
then 7<2. Therefore, o is ind"eed" a pseud.obase for (X,r), so that (X;r) is a Baire space since (X, o) is a Baire space. Thns (X,f) is a generating Baire space as desired".
3. k.Baire
ff
each V yrp,tle
spaces
Ltet k be a cardinal number larger than l{0. The topological space (X,9') is a hBai,re spa,ce if the intersection of fewer than /c dense open subsets of X is dense in .x. A subset H of. X is a G6,,,set if there is a set Qt c{ suchthat p/l
V. Products of Bairc
PnoposrtroN 5.14. The folloraing
e$Se
4{re
i'j't], 
,:
lprt p: nes€ .'i;
:."
a,?'a
spaccs
egui'oalent
OD
for
a topol'ogi'cal, space
X:
(i) X as a kBai,ro spa,ce. (ii) Eaery nonem,pty opten su,bset of X i,s of second, ltcategorg. {n\ Ihe oom,Tilement of any set oJ fi,rst hcatagorg 'in X 'is elense in X. PnoposnroN 5.15. Let X be a ltBai,ro Trspace u'ith, tto isol,ated, points, and,let T be aG5,*subset of X toi.th ly
*r. Then there are eractllyN, strbsefs of B whiclt, haue carddnnl'ilg I,ess than *r. Conor,r,ann 5.L7. Let r1t ba u +t'onl''imit' ot'd"i+tal' and' let (X,g) be a' togtologi,cal, space ai,lh iTl:*r. Then there are etnactl'g N,, 6;srs?/bsets o.f X. ft is necessary to require that p be a nonlimit orilinal in hoposition card,inali,ty
$*eF*
,,]tr
;;; f'4: x6
5.16 since a set of card.inality N. has exactl.r s*, ,, Eubsets of carclinality ' less than l*.. Tnnonpm "c.78. Let p be a nonzel'o orcl'inal, uhliclr' 'is not a li,mit ordi,'nal. Let (X,{) be a Hausd,orff space of earili,nalitg ** utlr,ich, hns no'isolated, po,ints. Btt'ltposa either I
(i) X zs an t\nBaire sp{rce wi,th l,{l : Nr, ot (ii) eoery first *rcategory subset of X is nowhere
.:i b,.!ss .i:.. ,:.,i
, .
Ttas a pseud,obase
fi
with l9l
:Sr.
d.ense
'i,t?,
X
ancl
X
Buppose furtkor tluat there enists a fi,rst category sttbset A oJ X2 sttch, X{*u Xl XA*i,s of fi,rst category i,n X} and, X'{*, Xl X A'i,s of first category i,n X\ are of fi,rst category 'in X. Th,eto tltere eu'ists a tlense subspace Y of X whi,clr, 'is an *,rBa'ire space and, ushose sgltcu'e i,s of first
that
ryf" &8BtI
.,**i FPqF
s*ts lg.ss
:':*
sryq
category i,tr, i.tself.
Proof. Let,Qt : {[]e{ I Uis nonernpty and contained. in the closnre of some first N,category subset of X). If Qt :6,let { : g and. $r :6. On the othei hand" if. q/ + O, clefine f and, 7/r as hollows. Iret D be the set of all pairwise d"isjoint subcollections of. t7/, partially orclered. by set inclusion. By Zorn's lemrna, D has a maximal elernent, which is ryhat ,we take fot {. Now clefine ltr : U{Ul U€lr'}. Let Vybe the subspace
*ql+ ;wd : .:.
topology on 7// inheritecl frolnt ,7. l;,et ae{'. }hen there exists a collection {lI"} of fewer than N" nowhere densesubsets of f, suchthat [rc cl([JrY,). For each a, let rY"iU)
e
: ]fonU. Then lf,(U) is a nowhere clense sulrset of X. Suppose there is an elemetrt r of U which is not in cl[U]f"(U)]. Then there is an open set Iz containing r such that f'^[Ufr"tUtl :O. Thnsr for each a, 7n1tr.(U) : yn;lr onil : 6 so that Q oaln {1_.;lr") : O. Therefore,
w+1
esic
wf #gr.
a
a
5
C:
Baire
66
,rd
cl{Uil")
uc
uA
el
spaces
which contradicts U being a subset ot
cl([_lrv"o). Therefore,
[U lr"(U)]. Nov'
"
,n
: u{Ui Uer'}c ll{ci[(JJr"(u)l I a,t'l
:l:: ,
 cl(tj{Vt"(a\
:
uev/l)
ct[U (U {x"( u)  u,
y}ll.
By Proposition 1.2, each [J {l[ U ,l'] is nowhere dense in X. There"(U)  fore, T : y (U {lf,( U)  U, "/.\l is of first l{rcategory in X and conseqnently in '/{, since 4tr is open in X. Note that 4/" c. cIT so that ? is a d.ense first ltrcategory subset of 7{. Bitce 7/r is an open subset of the NrBaire space X, it is an f{rBaire space. Clearly 71. cantains no isolated points. Bj Proposition 5.15, Wl : f{, since W is apen in X. Now 9"r. c{ so that i{y.l {lgl : Nr. Since l" is I[ausdorff, the complement of a single point is open. Therefore, l{*.l2rr** so th:art lgl:Sr. Define z  xclfr.antl let 7"be the subspace topology of z inherited rrom{. Let B be the first lt,categoly subset of Z. Assume that there exists a nonempty open subset v of z such that v c elsY c c158. Now Z is open in X, V is open in X, and. rS is of the first ftrcategory ii X. Therefore, VeQt.,Llso Vncl$/r:O so that VnTl:O for ali Ue /. Thus, /'v{Y} is an elernent of Zl which is strictly larger thanf ,which cont'radicts the maximality of 7". Therefore, eyery first l{rcategory subset of z is nowhere dense tn z. rt z * 6, t]nen z is a rlausd,orff space with cartlinality ltr, and contains no isolated. points. perhaps lf zl **,, llrat Z has a pseudotrase 4 such that l4l  l{e. First, we shall with the case x'here { + g and. Z * @. Iret t( be the set of all somewhere d.ense Gr,n,subsets of Nl.. Obviously, l#l2l{yl  f{"' sinee each G6,;.subset' of" l/' is associated with a corlection of fewer than elements of {rs, I,#l
and let
M : X{n, Xl XA, is of first categor;' in X}, Jf  X{r. Xl XA*'is of first category in X},
: A,tlM x (X")lut(r T)xMlvA, where A : {(rrn)l nuX}. Since X is llausdorff, / is closed" in X2. Assume that there exists a nonempty open set of X2 continued in /. Then there exists open seis Tz. and. F. in X such tlnat VrxV2c l. Therefore, C
7,
&;ti*:!l,jn;ti$,r';.;:i
,;4
and.
7, must be the same singleton point, but this contradicts the fact
a,.:ra
Y. Proclucts of Baire space$ €{cre, q
*
;
i t 1 ?,;
&*::ea. &ie
cfis d the wSate"d
YCf *ii:rgle
'drn* {hat : {'l r,S. ir;ra .5. '$e 9"'. FEICN
$qerv 5:*p&e€
;
F$er ,}F
&
DE
iF*'l
i$*x of '{ ***}'
f,
,,g,s
3
67
that X has no isolated points. Thus / is a nowhere dense subset of X2. By hypothesis 4, M, and.lf are of first category in X2. Therefore, 0{ x x(X?) and 6f) xlf are of first category in X2. Thus, C is of first category in X2. Define B : C olX x (x")lnl(x r) x Xl. Sirice B is contained in C, B is of first category in X2. We shall use transfinite induction to d.efine {n"l o
" 
Cro c.
(X

which is of first Nrcategory in X. Therefore, XC"s is of first l{rcategory in X. Now B*o C,onlX x (X  ?)l,on l(.f,  T) x Xf,o  C,oA6  f) Thus, X  B,s : (X C*r)vT which is of first l{rcategory in X. Therefore Z  B*s : (X  B,) oZ is of first l{rcategory in Z since Z is open in T. Now ZBr() #Z since Z is an l*rBaire space. Similarilyr ZB'o is of first N,category in Z. Now Z(B,roB"o1 : (ZB,o)v(ZBr0) is a first It,category subset of Z, and., hence, is nowhere d.ense in Z. 'Therefore, G:on(B*oaB*o) +g.Letgo,Gsn(BntnBr0). Note that {{r, n)  ne6f)} c B. Therefore, (.r0, Uo), (Uo, fio), (fry, ro), a,nd. (Uo, Uo) are all in B. Now let a
XLXIx(XT))" 0:x(xt):t
.
Therefore,
 n @ *uo B'F nB pnBa : l)l(11'B*)v(f B*P)v{1/r Bru)v(1/r B
E* As
L Yhen sf.lore, 3se
tact
o
P)
_ BU\l
is a first l{*category subset of "//r. Ilence, 7/rn[O"(B,unBrrnBrunBoul u F is a dense l{, Baire subspace of llr . Thtts, H oA l) tB, unn' aB, uo B' )l + A.
Baire
Let n
Now
ffo<
UlrT)
",
since
Z
UtV  B*t)u  fr

(Z
H
"ol1 5
u!
sPaces
(B * p^B'P oB a p^Bu P)l.
(B*unB*onBvprtBsl)
c (X?)'
Now
(\(B,B,rBqP^BvpnB*f aB*,oB*"\

Bai)v (Z Bu)v(Z

Bunlv (Z

B*")v(Z
 Br")l
i
':'i
is nowhere dense irt Z' lf.lnen 'i
is a first N"categoly subset of Z, and, hence, e..l1n (Biuan;l.nbnunBuonBn"'AB*"11is nonempty. I,ret Eobe an element
j
ot tff," *"t.
Tt : {n,la { b{r} and. Y, : {g.l o a t{r} which have thus been d.efineclbytransfiniteintluction, andlet Y : YtvTr. Clearly,Y is ad*1t" ,, subspace of X. Now Y, is a dense subset of 4/r atd eacb' Eointersects Y' ; Therefore, Y is an l.trBaire space. Ilet V be any open set in Yr. T1"1 I , ,l is a somewhere dense sulrset of Tr. Therefore, since Y, is dense inZ, V is Ltet'
;
:
somewherec1enseinZ.Thus,r/isofsecond{*categoryinZsot1rat7 isofsecond{,categoryinY,.Therefore,Y,isanN'BairesPace.Now
y:: Tow and I, : TnZ which makes T, and" I', open subsets of. I Therefore, Y is an ltrBaire space. By construction, Y2 c B. Sinc€ B
',
isoffirstcategoryin.k,,Y,isoffirstcategoryinX2.tr'ina1ly,sinc,eY2 is denso in X2, Y2 is of first category in itself. Tt Z :@ wetall uncLer part (i) of our hypothesis. T/'is then a rlense
:
X ancl the above proof suffices with only obvious alterations' : fi. l// 6 we fall und"er part {ii) of our hypothesis. z ]s then equal to X ancl the above proof suffices if we let B : C : AvU[ x(XM\]v u[(XLr)x N)vA. Conor,lanv 5.19. ,4sstlmi,ng the comtinuurn hypothesi,s, i'f (X,{) i's ,'
subspace of
],:
countable Hausd,orJf Bui,ro space of card,i'nal,i'tg *tt'ohi'ch has no i'sol'ated,lto,ints,thanthere d,oes not e*i,sts afirst categorg subset A of Xz sttch that ': Xln,Xl XAsi,s of first category i,n X\ and' X{neXlXA" ds of '1
a
secomd,
:.
'i fi,rst category'i,n X) are of first categor'31 in X. proof. Since X is lfausdorff, point complements are open. Therel f.ore, I{l}bir. Now each open set in x is associatecl with a sequence : :l{, S*ppose ftt. that so lgl of elements of the base. Thus,lgl<2$o there d.oes exist srrch an A. Then by Theorem 5.18 there exists a clense ;; subspace T of. x which is a Baire space and. whose square is of first :nTgo:I ::; in itself. Ttrerefore, by Theorem 1.L\ y would tre of first category in itself' " .,1
:
4. Product counterexamPles Tho next example [52] will illustrate that Theorems
'.'1i
1.11 ancl 5'9
cannot be proved without some restriction in the place of the countability '
V, Products of Baire
+l &e.n
**at j *cx. :t:
FSe
.T'
sF T is
r€
Tr
H*rs"
{I
:$ j.'
g{?te
i$**" ::i s"g*sl
r33u
3rs **pt f,*et Bs pJ
ryre .*1*.{]e i!: i'
€sFe !eslse
SirI E$€lI. :
*,*9 ,$ftr5
spaces
69
hypothesis. This 'was the first example of two Baire spaces s'hose ploduct is not a Baire space. Other examples use this kind. of example in their constluction. In this example it was necessary to use measule theory and the continuum hypothesis. This suggests that the prod"uct question might be a settheoretic type of question. Actuali;" Tall 164l points out thatt for oxtoby,s construction, it suffices to assume lVlartin's axiom (i'e., e1rery compact Ilausciorff space having the countable chain cond'ition is a 2NoBaire space) in place of the continuum hypothesis. \4re now give a brief description of Oxtoby's construction' ILet M ne tne field of L,ebesgue measurafule subsets of the unit interval, in J}{, and X the Stone space [27] associatecl If, the coilection of a1l nuilsets with the Boolean algebra MlN. X is a compact Ifausd"orff space with no Furtherisolated points. lxl :N, and X has a base of cartlinality t{t' oxtoby x' Ln' dense nowhere is x of subset first category more, "o""y version measule theoretic a using provecl tho following Lemma in 152] of Fubini's Theorem. I_,pmrru. 5.20. Ior X constructed, abotse, x2 conta'ins a subset a of first xcutegorE snch that x{u, xt xA,'is of first categorg i,n x} and' X' i'n category of are X} in XA* i,s of fi,rst category fi'rst {orXi Now using Theorem 5.18, one can conclude the existence of a clense Baire subspace of x rn'hose square is of first category in itself. lYe now give another example of a space satisfying the hypothesis of Theorem 5.18. Let Xobe the reals with the density topology (see t'he discussion after Theorem 3.5; also see [24], [61], 1651, ancl f?llforploperties of this space). The subset a of xoxxp in Theorem 5.18 can be eonstructecl in a rnanner similar to that which oxtoby used, in the proof of Lemma 5.20. tr'or a pseudobase having cardinality ltr (assuming the continuum hypothesis) take the usual ?"subsets of the reals having positive measure. Now let Tobe the d.ense subspace of xo which is given ty Theorem b.18. Then Y, is a Tysftsrtff Baire space having CCC whose uqorr* is of first category. An aclcled. property tlnt Yo has that Oxtoby's example did not haYe is that YD is hered.iaril5' 3"*. since x, is' \Yhite gives a moclification of this construction in [?2], in which he constructs a d,ense subspace z of. x, whose squale is of first category is, z and. which has the further property of being a sierpinski set; that Z that has countable intersection with eYery null set. From this it follows is hered,itarily Lindeliif (see [65]). A natural question now is whether there is such an. example which is a metric space. As for being hered.itarily Baire, the answer is no because
of Proposition 1.9. we will no$' show how Krom [34]
oxtoby's examp]e to show the existence of a metrizable Baire spaee whose squale is not a Baire spa'ce' First. we need. some new nota,tion. For a topological space (Xr{), let used"
Baire
spaces
:1st N>41 s(tz)=s(mf1) for eacion€l[and ff,trl 4:{{gl,f anci ( Ur, ..., a,) , {s. Xl s(1,) : fJt for 1< ? < rr1. tfofietth*t t0\, :
{(fJr,...,U,,)l Ur,...,Un, B} is a base for X. frfisRpM 5.2L. Ior ang togtologi,cal spaces X and, y, X xy sgtace i,f ancl, onl,y if E xY i,s a Bai,re sp&ce. B
i,s a Bai,re
Proof. For convenience of notation we will only show that (x,T) is a Baire space if and" on15r lf i is a Baire space. A similar technique can be used. on the general case. Using the notation of Theorem 3.16, tet gr(h): {f. S1&11 tor each (Ur,..., Un) , h, 111ai,..., U*)) : (Ur,..., U,o, U*r)' for so
me
U,r*1,
fij.
Define
a
onetoone corresponclence
a:
L)r@">6 by
o{(tlr: ..., Uo)) : (Ur, ...,
Un) .Define the correspond.ence 0: g*(fi)> g*(g) >9(#) as follows. Tor /< and f : (ar.,..., Un), h bt p(fi@) : (Uu._.., Un,flat10;1;. for 0, h d"tioe yLitt g(61rs"(9) and. y'6: 9(0)>9*(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 br(fr), yrfri), gli +g. h can tre shown that ff La,i, fi@)lt+d]'nno*, i is a Baire il 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 zerodimensional 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
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tba*
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I'NMRSITY
'.:r:
and STATE
1,:1: :,' :
Blacksburg, Yirginia 24061 USA
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