Comment.Math.Univ.Carolinae 40,4 1999 771{775 771
Pro ducts, the Baire category theorem,
and the axiom of dep endent choice
Horst Herrlich, Kyriakos Keremedis
Abstract. In ZF i.e., Zermelo-Fraenkel set theory without the Axiom of Choice the
following statements are shown to b e equivalent:
1 The axiom of dep endentchoice.
2 Pro ducts of compact Hausdor spaces are Baire.
3 Pro ducts of pseudo compact spaces are Baire.
4 Pro ducts of countably compact, regular spaces are Baire.
5 Pro ducts of regular-closed spaces are Baire.
6 Pro ducts of Cech-complete spaces are Baire.
7 Pro ducts of pseudo-complete spaces are Baire.
Keywords: axiom of dep endent choice, Baire category theorem, Baire space, count-
ably compact, pseudo compact, Cech-complete, regular-closed, pseudo-complete, pro d-
uct spaces
Classi cation : 03E25, 04A25, 54A35, 54B10, 54D30, 54E52
Concerning the status of the Baire category theorem for compact Hausdor
resp ectively Cech-complete spaces in ZF the following results are known:
Theorem 1 [1], [8] . Cech-complete spaces are Baire if and only if the axiom
of dep endentchoice holds.
Theorem 2 [7] . Compact Hausdor spaces are Baire if and only if the axiom
of dep endentmultiple choice holds.
Theorem 3 [3], [11] . Countable pro ducts of compact Hausdor spaces are
Baire if and only if the axiom of dep endentchoice holds.
The natural question asking for the set-theoretical status of the statement
\arbitrary pro ducts of compact Hausdor resp. Cech-complete spaces are Baire"
has b een left op en so far. The purp ose of this note is to close this gap. Recall:
De nitions. 1 A top ological space X is called Baire provided that in X the
intersection of any sequence of dense op en sets is dense.
1
2 A lter on a space is called regular provided that it has a closed base and an
op en base.
1
Filters on X are always supp osed to b e proper subsets of the p ower set of X .
772 H. Herrlich, K. Keremedis
3 A top ological space X is called regular-closed provided that X is regular and
any regular lter on X has a non-emptyintersection. See [10].
4 A collection B of non-empty op en sets of a top ological space X is called a
regular pseudo-base for X provided that B satis es the following conditions:
for each non-empty op en set A in X there exists some B 2Bwith cl B A,
if A is a non-empty op en subset of some B 2B, then A 2B.
5 A top ological space X is called pseudo-complete provided that it has a se-
quence B of regular pseudo-bases such that every regular lter on X , that
n
n2N
has a countable base and meets each B , has a non-emptyintersection. See [Ox] .
n
Such a sequence of regular pseudo-bases will b e called suitable for X .
Remark. Each compact Hausdor space is simultaneously
a countably compact and regular,
b pseudo compact,
c regular-closed,
d Cech-complete.
Moreover, each top ological space that satis es a , b , c or d is pseudo-
complete.
Theorem 4. The following conditions are equivalent:
1. The axiom of dep endentchoice.
2. Countable pro ducts of compact Hausdor spaces are Baire.
3. Pro ducts of compact Hausdor spaces are Baire.
4. Pro ducts of pseudo compact spaces are Baire.
5. Pro ducts of countably compact, regular spaces are Baire.
6. Pro ducts of regular-closed spaces are Baire.
7. Pro ducts of Cech-complete spaces are Baire.
8. Pro ducts of pseudo-complete spaces are Baire.
Proof: In view of the ab ove Remark, condition 8 implies the conditions 4 ,
5 , 6 , and 7 , and moreover, each of the latter conditions implies condition 3 .
Since the implication 3 2 holds trivially and the implication 2 1 holds
by Theorem 3, it remains to b e shown that condition 1 implies condition 8 .
Assume condition 1 to hold. Let X be a family of pseudo-complete
i
i2I
Q
spaces and let X = X be the corresp onding pro duct with pro jections
i
i2I
: X ! X .
i i
Case 1: X = ;.
Then X is Baire.
Case 2: X 6= ;.
Let x = x be a xed element of X . Let D be a sequence of dense
n
i
i2I n2N
op en subsets of X and let B b e a non-empty op en subset of X . Consider the set
Y of all quadruples
n; F ; B ; B
i i
i2F i2F
Pro ducts, the Baire category theorem, and theaxiom ofdep endentchoice 773
consisting of
a a natural number n,
b a nite subset F of I ,
c a family B of non-empty op en subsets B of X ,
i i i
i2F
n
d a family B of suitable sequences B of regular pseudo-bases
i
i2F
n2N
i
for X ,
i
sub ject to the following conditions:
T
1
[B ] B \ D , e
n
i
i2F
i
m
f B 2B for each i 2 F and each m n.
i
i
m
The fact that each B is a regular pseudo-base implies that Y is non-empty.
i
Consider further the relation de ned on Y by:
If y = n; F ; B ; B
i i
i2F i2F
~ ~ ~
and y~ = n;~ F; B ; B
i i ~ ~
i2F i2F
then yy~ i the following conditions are satis ed:
n +1 =n ~,
~
F F ,
~
cl B B for each i 2 F ,
i i
X
i
~
B = B for each i 2 F .
i i
m
is a regular pseudo-base implies that for each The fact that each B
i
y 2 Y there exists some y~ 2 Y with yy~. Thus condition 1 guarantees
the existence of a sequence y in Y with y y for each n, and y =
n n n
n+1
n2N
S
n
; B . The set F = F is, by condition 1 , as a n; F ; B
n n
i
i2F i2F
n n n2N
i
countable union of nite sets at most countable. For each i 2 F , consider
n
is a n = min fn 2 N j i 2 F g. Then for each i 2 F the sequence B
n
i
nn
i
i
n m
for all m n and all n m. 2 B base for a regular lter on X with B
i i
i i
Thus pseudo-completeness of the X 's implies that, for each i 2 F , the set
i
T
n
B = B is non-empty. By countabilityof F and the fact that 1 implies
i
nn
i
i
Q
the axiom of countable choice, there exists an element b in B . Thus
i i
i2F
i2F
n
T
b ; if i 2 F
i
the p oint y , de ned by y = b elongs to B \ D .
n
i i
i2I
n2N
x ; if i 2 I nF ;
i
T
Consequently D is dense in X .
n
n2N
Remarks. 1 That Case 1 in the ab ove pro of may o ccur even if all the X 's are
i
non-empty compact Hausdor spaces is shown by the mo del N 15 in [12]. Thus
in ZF the statement
* Pro ducts of non-empty compact Hausdor spaces are non-empty and Baire
is prop erly stronger than the axiom of dep endentchoice.
774 H. Herrlich, K. Keremedis
2 By Theorem 4 each of the statement 1 { 8 is a theorem in ZFC i.e.,
Zermelo-Fraenkel set theory including the axiom of choice . In particular the
following are known:
a Complete metric spaces are Baire. See Hausdor [9].
b Pro ducts of completely metrizable spaces are Baire. See Bourbaki [2].
c Compact Hausdor spaces are Baire. See R.L. Mo ore [13].
d Countably Cech-complete spaces are Baire. See Cech [4] and Gold-
blatt [8].
e Pro ducts of Cech-complete spaces are Baire. See Oxtoby [14].
f Countably compact, regular spaces are Baire. See Colmez [5].
g Pseudo compact spaces are Baire. See Colmez [5].
h Pseudo-complete spaces are Baire. See Oxtoby [14].
Observe that in ZFC none of the following prop erties is closed under the formation
of pro ducts:
Baire see, e.g., [6, 3.9.J.] ,
pseudo compact see, e.g., [6, Example 3.10.19.] ,
countably compact, regular see, e.g., [6, Example 3.10.19.] ,
regular-closed see [15] ,
Cech-complete see, e.g., [6, 3.9.D. a ] .
Observe further that in ZFC all the ab ove results follow from Oxtoby's [14]
results h ab ove and
i Pro ducts of pseudo-complete spaces are pseudo-complete.
But, whereas h holds in ZF + DC = the axiom of dep endent choice , the
result i seems to require far stronger selection principles. Thus each of the
results 3 { 8 , considered as a theorem in ZF + DC, is new.
References
[1] Blaire C.E., The Baire Category Theorem implies the principle of dependent choice, Bull.
Acad. Math. Astronom. Phys. 25 1977 , 933{934.
[2] Bourbaki N., Topologie g en erale,ch. IX., Paris, 1948.
[3] Brunner N., Kategoriesatze und Multiples Auswahlaxiom, Zeitschr. f. Math. Logik und
Grundlagen d. Math. 29 1983 , 435{443.
[4] Cech E., On bicompact spaces, Ann. Math. 38 1937 , 823{844.
[5] Colmez J., Sur les espaces pr ecompacts, C.R. Acad. Paris 234 1952 , 1019{1021.
[6] Engelking R., General Topology, Heldermann Verlag, 1989.
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[9] Hausdor F., Grundzuge der Mengenlehre, Leipzig, 1914.
[10] Herrlich H., T -Abgeschlossenheit und T -Minimalitat, Math. Z. 88 1965 , 285{294.
[11] Herrlich H., Keremedis K., The Baire Category Theorem and Choice, to app ear in Top ology
Appl.
[12] Howard P., Rubin J.E., Consequences of the axiom of choice, AMS Math. Surveys and
Monographs 59, 1998.
Pro ducts, the Baire category theorem, and theaxiom ofdep endentchoice 775
[13] Mo ore R.L., An extension of the theorem that no countable point set is perfect, Pro c. Nat.
Acad. Sci. USA 10 1924 , 168{170.
[14] Oxtoby J.C., Cartesian products of Baire spaces,Fund. Math. 49 1961 , 157{166.
[15] Pettey D.H., Products of regular-closed spaces,Top ology Appl. 14 1982 , 189{199.
Department of Mathematics, University of Bremen, Postfach, 28334 Bremen,
Germany
E-mail : [email protected]
Department of Mathematics, University of the Aegean, Karlovasi, 83200 Samos,
Greece
E-mail : [email protected]
Received Octob er 19, 1998