Topology Proceedings 42 (2013) Pp. 275-297: Wallman Compactifications and Tychonoff's Compactness Theorem in {\Sf

$Topology Proceedings 42 (2013) Pp. 275-297: Wallman Compactifications and Tychonoff's Compactness Theorem in {\Sf$

Volume 42, 2013 Pages 275–297

http://topology.auburn.edu/tp/

Wallman Compactifications and Tychonoff’s Compactness Theorem in ZF

by Kyriakos Keremedis and Eleftherios Tachtsis

Electronically published on January 26, 2013

Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT ⃝c by Topology Proceedings. All rights reserved. http://topology.auburn.edu/tp/ TOPOLOGY PROCEEDINGS Volume 42 (2013) Pages 275-297 E-Published on January 26, 2013

WALLMAN COMPACTIFICATIONS AND TYCHONOFF’S COMPACTNESS THEOREM IN ZF

KYRIAKOS KEREMEDIS AND ELEFTHERIOS TACHTSIS

Abstract. We show that the following statements are pairwise equivalent in ZF: (1) The axiom of choice (AC). (2) For every T1 space (X,T ) and every T1 base C for X, the set W(X, C) of all C-ultrafilters when endowed with the Wallman topology TW(X,C) (definitions are provided in section 1) is a compactification of X. (3) For every T1 space (X,T ), for every T1 base C of X, every filter base G ⊂ C extends to a C-ultrafilter F. (4) “For every T1 space (X,T ), W(X) (here C = K(X), the family of all closed subsets of X) is a compactification of X” and { ∈ } “For∏ every family∏ (Xi,Ti): i ω of compact T1 spaces, W( Xi) and W(Xi) are topologically homeomorphic.” i∈ω i∈ω We also show that “ For every T1 space (X,T ), W(X) is a compactification of X” implies that every infinite set has a countably infinite subset. In addition, we show that “ For every T1 space (X,T ), W(X) is a compactification of X” if and only if CFE1 (= every filter base G of closed subsets of a T1 space (X,T ) extends to a closed ultrafilter F).

1. Notation and Terminology { ∈ } ∏Let (Xi,Ti): i I be a family of topological spaces and let X = Xi be their Tychonoﬀ product. A closed subset F of X is called basic i∈I closed in case ∪{ −1 ∈ } ∈ <ω ∈ c ∈ F = πq (Fq): q Q where Q [I] and for all q Q, Fq Tq.

2010 Mathematics Subject Classification. Primary 03E25; Secondary 03E35, 54B10, 54D30. Key words and phrases. axiom of choice, compactness, closed ultrafilters, Ty- chonoff’s compactness theorem. ⃝c 2013 Topology Proceedings. 275 276 K. KEREMEDIS AND E. TACHTSIS

We shall denote the collection of all basic closed subsets of X by C(X). C(X) is a base for the closed subsets of X (every closed subset of X can be expressed as an intersection of members of C(X)). A closed set F of X is∏ called restricted closed ∏if there is a finite set Q ⊆ I such that F = V × Xi, with V ⊂ YQ = Xi a closed set. Note that for a given i∈Qc i∈Q restricted closed set F in X, there is not a unique set Q satisfying the aforementioned condition. (Let F , V , and Q be as in the latter definition, | \ | ≥ ∈ \ and (for our convenience)∏ suppose that I Q 2 and let i0 I Q. ′ × ′ ∪ { } ′ × Then F = V Xi, where Q = Q i0 and V = V Xi0 ). i∈(Q′)c However, it is fairly easy to see that to every nonempty proper restricted closed subset F of X, there corresponds a finite set QF ⊆ I which is the ⊆-smallest set Q in [I]<ω with respect to the above condition on Q. (If F is a nonempty (proper) restricted closed set in X, then the ⊆ ⊆ -smallest set QF I can be defined∏ so that there is a closed set VF × ∩{ ∈ } in YQF such that F = VF Xi, VF = Zj : j J , where c i∈(QF ) ∪{ −1 ∈ } ⊂ ∈ ∈ Zj = πq (Gqj): q QF YQF and for all q QF , for all j J, c ∈ ∈ ∈ Gqj Tq, and for each q QF , there exists an index j ∏J such that <ω Gqj ≠ ∅,Xq. Then any Q ∈ [I] for which F = V × Xi with i∈(Q)c V closed in YQ is such that Q ⊇ QF ; recall that V is expressible as an ′ ∪{ −1 ′ ∈ } ∈ ′ intersection of basic closed subsets Zj = πq (Gqj): q Q , j J , of ∈ ∈ ′ ′ YQ. Then discard those q Q, each being such that for every j J , Gqj is either empty or all of Xq.) The set QF is called the set of restricted coordinates of F . The collection of all restricted closed subsets of X shall be denoted by CR(X). Clearly, C(X) is closed under finite unions but not closed under finite intersections. However, the collection C(X) consisting in all sets of the form

∩{∪{ −1 ∈ } ∈ } (1.1) G = πq (Fqi): q Qi : i n

∈ N ∈ ∈ <ω ∈ c ∈ where n and for all i n, Qi [I] , and for all q Qi, Fqi Tq, or, equivalently, ∏ ∪{∩{ ∈ } ∈ { −1 ∈ }} (1.2) G = f(i): i n : f πq (Fqi): q Qi , i∈n

is easily seen to be a base for the closed subsets of X, closed under finite unions and finite intersections. Likewise, CR(X) is a larger base for the closed subsets of X which is also closed under finite unions and finite intersections. WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 277 ∏ For every S ⊂ I, pS : X → YS = Xi will denote the projection of i∈S X onto YS. In particular, if S = {i} for some i ∈ I, then we will denote the projection pS by πi. C1(X) denotes the collection of all 1-basic closed C { −1 ∈ c ∈ } subsets of X, i.e., 1(X) = πi (F ): i I,F Ti . Let (X,T ) be a topological space. X is compact if every open cover U of X has a finite subcover V. Equivalently, X is compact if and only if for every family∩ G of closed subsets of X having the finite intersection property (fip) G ̸= ∅. Let X be a nonempty set and let E be a collection of subsets of X which is closed under finite intersections. A nonempty subcollection F of E\{∅} is an E-filter if and only if (i) if F1, F2 ∈ F, then F1 ∩ F2 ∈ F. (ii) if F ∈ F, F ′ ∈ E and F ⊆ F ′, then F ′ ∈ F. If E = P(X), then an E-filter is called a filter on X. If E is the collection of all closed subsets of a topological space, then we say that F is a closed filter. A nonempty collection H ⊆ E \ {∅} is called an E-filter base if, for ∈ H ∈ H ⊆ ∩ every H1,H2 , there is an H3 ∩ such that H3 H1 H2. A filter base H ⊆ E \ {∅} is called free if H = ∅. A maximal, with respect to inclusion, E-filter is called an E-ultrafilter. An E-filter F is called countably closed if it satisfies the condition: If {Fi : i ∈ ω} ⊆ F, then ∩{Fi : i ∈ ω} ∈ F. An E-filter F is called countably prime if it satisfies the condition: If {Fi : i ∈ ω} ⊂ E and ∪{Fi : i ∈ ω} ∈ F, then there exists i ∈ ω such that Fi ∈ F. Let (X,T ) be a T1 space and let B be a base for the closed subsets of X. Then B is called a T 1 base for X if it satisfies the following. (i) ∅ ∈ B. (ii) If B1,B2 ∈ B, then B1 ∩ B2 ∈ B and B1 ∪ B2 ∈ B. (iii) If x∈ / B ∈ B, then there is Bx ∈ B such that x ∈ Bx and Bx ∩ B = ∅. { ∈ } C If (Xi,Ti): i I is a family of T1 topological∏ spaces, then R(X) and C(X) are T1 bases for the product X = Xi. Note that C(X) is not a i∈I T1 base for∏X. If X = Xi is a product of topological spaces and F is an E-ultrafilter, i∈I where C(X) ⊂ E ⊆ K(X) (= the family of all closed subsets of X), then for every i ∈ I, Fi denotes the family of all closed subsets of Xi whose inverse image under πi is a member of F, i.e.,

F { ⊆ c ∈ −1 ∈ F} (1.3) i = F Xi : F Ti, πi (F ) . 278 K. KEREMEDIS AND E. TACHTSIS

Let (X,T ) be a T1 topological space and let C be a T1 base for X. W(X, C) denotes the set {F ⊂ C : F is a C-ultrafilter}. In particular, for C = K(X), we shall denote W(X, K(X)) by W(X). The topology TW(X,C) on W(X, C), having as a base for the closed sets the family B = {A∗ : A ∈ C},A∗ = {F ∈ W(X, C): A ∈ F}, is called the Wallman topology corresponding to the base C. Since for every A ∈ C, (A∗)c = {F ∈ W(X, C): ∃F ∈ F such that F ∩ A = ∅}, it follows that (1.4) U = {U ∗ : U c ∈ C} and U ∗ = {F ∈ W(X, C): ∃ F ∈ F such that F ⊆ U} is a base for W(X, C). In the sequel, W(X, C) will denote the topological space (W(X, C),TW(X,C)). We shall be referring to W(X, C) as the Wallman space of X corresponding to the base C. AT1-compactification of X is a compact T1 space (Y,Q) such that X embeds in Y as a dense subspace. We recall (See [14, Theorem IV.3, p. 138]) that, in ZFC, a Wallman compactification of (X,T ) is the topological space W(X, C). If (X,T ) and (Y,P ) are topological spaces and f : X → Y is a 1 : 1, onto continuous and open mapping, then we will write X ≃ Y and say that X and Y are homeomorphic. BPI: Every Boolean algebra has a prime ideal. (It is known that BPI is equivalent in ZF to each one of the statements: “The Tychonoff product X of compact T2 spaces is compact” (see [15]) and “2 is compact for every infinite set X” (see [13]).) CFE (Axiom of Closed Filter Extendability): For every topological space (X,T ), every filter base G of closed subsets of X extends to a closed ultrafilter F. CFE1: CFE restricted to T1 spaces. { TCT (Tychonoff Compactness Theorem): For every family (Xi∏,Ti): i ∈ I} of compact topological spaces, the Tychonoff product X = Xi i∈I is compact. (It is known that TCT if and only if TCT restricted to T1 spaces (see [8]).) A { ∈ } AC(I): For every∪ family = Ai : i I of nonempty sets there exists a function f : A → A such that f(Ai) ∈ Ai for all i ∈ I. AC (Axiom of Choice): For every set I, AC(I). ACfin: AC restricted to families of nonempty finite sets. CAC (Countable Axiom of Choice): AC(ω). It is known (see [7]) that CAC is equivalent to its partial version PCAC; i.e., every countably infinite family of nonempty sets has an infinite subfamily with a choice function. WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 279

2. Introduction and Some Known Results If (X,T ) is a topological space, then one cannot prove the existence of a closed ultraﬁlter of X in ZF. In fact, Horst Herrlich [3] has shown the following. Theorem 2.1. Every topological space has a closed ultraﬁlter if and only if AC.

However, if (X,T ) is a T1 space, then one can always prove in ZF the existence of a closed ultraﬁlter. Indeed, for every x ∈ X, (2.1) F(x) = {F ⊆ X : x ∈ F,F c ∈ T }

is easily seen to be a closed ultrafilter of X. More generally, if C is a T1 base for X, then F(x) = {F ∈ C : x ∈ F } is a C-ultrafilter. (This follows from condition (iii) in the definition of a T1 base; see section 1.) It follows that we can rephrase CFE1 as “every free filter base G of closed subsets of a T1 space (X,T ) extends to a closed ultrafilter F” and “For every T1 space (X,T ), for every T1 base C of X, every filter base G ⊂ C extends to a C-ultrafilter F” as “ for every T1 space (X,T ) for every T1 base C of X, every free filter base G ⊂ C extends to a C-ultrafilter F.” CFE was introduced in [6] where Herrlich and Juris Stepr¯ansestablished the following. Theorem 2.2. AC if and only if CFE + CAC. They asked whether CFE implies CAC . In [10, Theorem 2], it was shown that the answer to this question is in the affirmative. If the readers go through the proof of Theorem 2.2 as is given in [6, Proposition 2, p. 700], they will realize that they can use the same proof in order to establish the following.

Theorem 2.3. AC if and only if CFE1 + CAC. Unlike the case of Theorem 2.2, one cannot use the proof of Theorem 2 in [10] to show that CFE1 implies CAC. The space involved in that proof satisﬁes the T0 axiom but not the T1. In fact, this is an intriguing open problem we shall be concerned with in this paper.

Theorem 2.4. (i) AC implies “For every T1 space (X,T ) and every T1 base C for X, W(X, C) is a compactification of X” implies “For every T1 space (X,T ), W(X) is a compactification of X.” (ii) For every T1 space (X,T ) and for every T1 base C of X, every filter base G ⊂ C extends to a C-ultrafilter F if and only if for every T1 space (X,T ) and every T1 base C for X, W(X, C) is a compactification of X. 280 K. KEREMEDIS AND E. TACHTSIS

(iii) CFE1 if and only if “For every T1 space (X,T ), W(X) is a com- pactiﬁcation of X.” (iv) CFE1 implies BPI implies ACfin. Proof. (i) For the ﬁrst implication, see the proof of Theorem IV.3, p. 138, in [14]. The proof of the second implication is straightforward.

(ii) (→) Let (X,T ) be a T1 space and let C be a T1 base for X. To W C G { ∗ ∈ } prove that (X, ) is compact, we start with a family = Gi : i I , where for all i ∈ I, Gi ∈ C, having the fip. Then H = {Gi : i ∈ I} ⊂ C is a family with the fip. By our hypothesis, H extends to a C-ultrafilter F. F ∈ ∩{G∗ ∈ } ̸ ∅ W C Clearly, i : i I = , and consequently (X, ) is compact. In addition to being compact, see [14, Theorem IV.3, p. 138], W(X, C) is T1 and the mapping (2.2) φ : X → W(X, C), φ(x) = F(x), where F(x) = {F ∈ C : x ∈ F }, is a topological embedding such that φ(X) = W(X, C). Thus, W(X, C) is a T1 compactification of X.

(ii) (←) Let (X,T ) be a T1 space, C a T1 base for X, and H a free filter base of C. Clearly, {H∗ : H ∈ H} is a family of closed subsets of W(X,C) having the fip. Hence, by our hypothesis, K = ∩{H∗ : H ∈ H} ̸= ∅. Clearly, any F ∈ K extends H. (iii) Follow the proof of (i) with K(X) in place of C. (iv) The first implication is straightforward and the second is well known; see [7]. At this point, one may ask the following questions.

Question 2.5. Does “ For every T1 space (X,T ), W(X) is a compactiﬁ- cation of X” imply AC ?

Question 2.6. Does “ For every T1 space (X,T ) and every T1 base C for X, W(X, C) is a compactiﬁcation of X” imply AC ?

Question 2.7. Does “ For every T1 space (X,T ), W(X) is a compactiﬁ- cation of X” imply “ For every T1 space (X,T ) and every T1 base C for X, W(X, C) is a compactiﬁcation of X”? Taking into consideration Theorem 2.4 and Theorem 2.3, Question 2.5 reduces to the one introduced after the statement of Theorem 2.3. Namely,

Question 2.8. Does CFE1 imply CAC ? With regard to Question 2.6, we prove in Theorem 4.1 that the answer is in the aﬃrmative. Hence, Question 2.7 is simply a rewording of Question 2.5. WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 281

Regarding Question 2.8, the best we could get is stated in Theorem 4.3. Namely, CFE1 implies every infinite set has a countably infinite subset. We conjecture that the answer to Question 2.8 is in the negative. Before we set out proving our results, let us list a few results which we are going to use in the sequel. { ∈ } Proposition 2.9 ([9])∏ . Let (Xi,Ti): i I be a family of topological spaces and let X = Xi be their product. Then the following hold. i∈I (i) A closed subset F of X is restricted closed if and only if there ⊆ ∩{∪{ −1 ∈ } exists a finite set Q I such that F = πq (Fjq): q Q : ∈ } ∈ ∈ c ∈ j J , where for j J and q Q, Fjq Tq. (ii) The union (intersection) of finitely many restricted closed sets is a restricted closed set. In particular, if each Xi is T1, then CR(X) is a T1 base for X. (iii) Every closed subset of X can be expressed as an intersection of fewer than |[I]<ω|-many restricted closed sets. Proposition 2.10 (ZF). (i) The product of finitely many compact spaces is compact. (ii) Let (X,T ) be a topological space and let (Y,Q) be a compact space. Then the projection πX : X × Y → X of X × Y onto X is a closed map. (iii) Let (X,T ) be a topological space and B a base for X. Then X is compact if and only if every open cover U ⊂ B has a finite subcover if and only if every family H ⊂ {Bc : B ∈ B} having the fip has a nonempty intersection. Proof. The proof of (i) is well known and it is demonstrated in any stan- dard textbook, such as [12] and [18]. For the proof of (ii), see [12, Exercise 8, p. 172]. Finally, the proof of (iii) is well known and straightforward. Proposition 2.11. (i) AC if and only if TCT. (ii) TCT restricted to countable families of compact topological spaces implies CAC. Proof. (i) TCT → AC has been proved in [8]. There is a plethora of proofs of the implication AC → TCT; see [9, Theorem 2.1] for a recent one. In fact, in every book of general topology there is such a proof. The following proof is the shortest one attributing the necessary appreciation to CFE1. { ∈ } Fix∏ (Xi,Ti): i I a family of compact T1 spaces. We show that X = Xi is compact. Fix, by Proposition 2.10(iii), i∈I H {∪{ −1 ∈ } ∈ } ⊂ C (2.3) = πq (Fjq): q Qj : j J (X), ∈ ∈ <ω ∈ ∈ c ∈ where for all j J, Qj [I] , and for all j J and q Qj, Fjq Tq, a family with the fip and, by AC, let F be a closed ultrafilter extending H. 282 K. KEREMEDIS AND E. TACHTSIS ∩ For every i ∈ I, let Ai = Fi where Fi is given by (1.3). Since Xi ∈ Fi and Xi is compact, it follows that Ai ≠ ∅. By AC, fix ai ∈ Ai, i ∈ I, and let a ∈ X satisfy a(i) = ai for all i ∈ I. Since F is maximal, it follows that, for every j ∈ J, there is q ∈ Q with π−1(F ) ∈ F. Hence, for j j qj ∩ jq every j ∈ J, a ∈ π−1(F ), and consequently a ∈ H and X is compact qj jq as required. (ii) Use the proof of TCT → AC given in [8].

Remark 2.12. Regarding Question 2.8, we may conclude from the proof of Proposition 2.11 that if CFE1 implies for every i ∈ I, Fi is a closed ultrafilter, as is the case when F is a C(X)-ultrafilter or a CR(X)-ultrafilter (see [9, Theorem 3.2]), then CFE1 is equivalent to AC. In the spirit of Proposition 2.11 we could not resist giving a very simple and notably short proof of the following characterization of the Boolean Prime Ideal Theorem (BPI) provided by Eric Schechter [16]. Proposition 2.13. BPI if and only if the Tychonoff product of spaces each endowed with the cofinite topology is compact.

Proof. (→) Let {(Xi,Ti): i ∈ I} be a family of spaces such that Ti is the cofinite topology on Xi for every i ∈ I. Let X be its Tychonoff product. If X = ∅, then there is nothing to show. So let x = (xi)i∈I ∈ X. Let H be given by (2.3), and, by∩ BPI, let F be an ultrafilter of X including H. For every i ∈ I, let Ai = Fi, where Fi is given by (1.3). Clearly, Ai ≠ ∅ for all i ∈ I. Furthermore, as Ai is closed in Xi and Ti is the cofinite topology, we have that either Ai = Xi or Ai is finite. In the second case, it easily follows from the maximality of F that Ai is a singleton, say Ai = {ai}. Now define a function f ∈ X by requiring f(i) = xi if Ai = Xi, and f(i) = ai∩otherwise. As in the proof of Proposition 2.11, one may verify that f ∈ H ̸= ∅. Thus, X is compact as required. (←) The hypothesis readily implies the equivalent form of BPI “ for every infinite set X, 2X is compact.”

3. Some Preliminary Results

Proposition 3.1. Let (X,T ) be a T1 space and let C be a T1 base for X. If X has no free C-ultraﬁlters, then X ≃ W(X, C). In particular, if X is compact, then X ≃ W(X, C) ≃ W(X). Proof. The proof follows at once from the observation that W(X, C) = {F(x): x ∈ X}, where F(x) = {F ∈ C : x ∈ F }, and the proof of Theorem IV.3, p. 138, in [14]. (The function φ : X → W(X, C), φ(x) = F(x), is an onto topological embedding.) WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 283

Remark 3.2. (i) If T is the discrete topology on the infinite set X and X ≃ W(X), then X has no free ultrafilters. (If h : X → W(X) is a homeomorphism and F is a free ultrafilter of X, then F = h(x) for some x ∈ X. Since {x} ∈ T , it follows that {F} is open in W(X). Hence, there exists A ⊂ X with (A∗)c = {F}. Clearly, A ≠ X, so let y ∈ Ac. Since F is free, {y} ∈/ F. Consider the ultrafilter G = {G ⊂ X : y ∈ G} of X. Since {y} ∩ A = ∅, G ∈ (A∗)c and G ̸= F. This is a contradiction; hence, X has no free ultrafilters.) (ii) Let (X,T ) be a T1 space and h : X → W(X) be a homeomorphism. If F is a free closed ultrafilter of X, then X has a countably infinite subset. Indeed, F = h(x1) for some unique x1 ∈ X and by a straightforward induction, there exists (unique) xn+1 ∈ X\{x1, ..., xn} F { ⊂ c ∈ ∈ } such that h(xn+1) = xn = F X : F T and xn F . ∏ Clearly, in products X = Xi, projections of members of C(X) are i∈I closed sets. We show next, as expected, that the members of C(X) share the same property. This property of the base C(X) turns out to be very useful in the sequel. { ∈ } Proposition∏ 3.3. Let (Xi,Ti): i I be a family of topological spaces and let X = Xi be their Tychonoff product. For every F ∈ C(X) and i∈I for every i ∈ I, πi(F ) is a closed set. Proof. Fix F ∈ C(X). By (1.2), express F as ∏ ∪{∩{ ≤ } ∈ { −1 ∈ }} F = f(v): v n : f Y = πq (Fqv): q Qv . v≤n

Since |Y | < ℵ0 and for all f ∈ Y , πi(∩{f(v): v ≤ n}) is a closed set, it follows that πi(F ) is a closed set as required.

Proposition 3.4. Let {(Xi,Ti): i ∈ I} be a family of topological spaces with Tychonoff product X ≠ ∅ and let F and H be C(X)-ultrafilters. Then (i) F = F ∩ C1(X) is a maximal subfamily of C1(X) with the fip, and for every i ∈ I, Fi = πi(F) = {πi(K): K ∈ F} is a closed ultrafilter, where Fi is given by (1.3). Furthermore, the C(X)-filter G generated by F coincides with F. In particular, if F ̸= H, then there exists i ∈ I with Fi ≠ Hi, where for all i ∈ I, Fi and Hi are given by (1.3). (ii) If, for every i ∈ I, Fi is a closed ultrafilter of Xi, then the C(X)- F { −1 ∈ ∈ F } filter generated by πi (F ): i I,F i is a C(X)-ultrafilter.

Proof. (i) F is maximal with the ﬁp and πi(F) = {πi(K): K ∈ F} is a closed ultraﬁlter follow at once from [9, Theorem 3.2(i)]. (Note that 284 K. KEREMEDIS AND E. TACHTSIS

the compactness of the spaces (Xi,Ti) is not used in Theorem 3.2(i); F = (F ∩ C(X)) ∩ C1(X)) and F ∩ C(X) is a maximal subfamily of C(X) with the fip.) Fi = πi(F) follows from Proposition 3.3 and the fact that F is a C(X)-filter. Regarding the equality F = G, the inclusion G ⊆ F is obvious. To see F ⊆ G, fix F ∈ F and, by (1.1), express F as

(3.1) F = ∩{Zi : i ∈ n}, ∈ N ∈ ∪{ −1 ∈ } ∈ <ω where n ; for all i n, Zi = πq (Fqi): q Qi , Qi [I] ; and ∈ c ∈ F ∈ for all q Qi, Fqi Tq. By the maximality of , fix, for every i n, a q ∈ Q with π−1(F ) ∈ F and let G = ∩{π−1(F ): i ∈ n}. As i i qi qi qi qi F ⊇ G ∈ G, we see that F ∈ G and F ⊆ G as required. (ii) Fix F ∈ C(X) such that {F }∪F has the fip. We show that F ∈ F. Express F in the form of (3.1). Clearly, for every i ∈ n, {Zi}∪F has the fip. Hence, for every i ∈ n, there is a q ∈ Q such that π−1(F ) ∈ F . i i qi qi i (If not, then there is an i ∈ n such that for every q ∈ Qi, there exists, by the maximality of Fi, a closed subset Bq of Xq such that Bq ∈ Fq and −1 ∩ −1 ∅ ∩ ∅ ∩{ −1 ∈ } ∈ πq (Bq) πq (Fqi) = . Hence, K Zi = ,K = πq (Bq): q Qi F, contradicting the fip of {Z }∪F). Since F ⊇ ∩{π−1(F ): i ∈ n} ∈ F, i qi qi it follows that F ∈ F as required. In the next proposition we extract the essence of the proof of Proposi- tion 2.11(i) in order to use it in the rest of the paper.

Proposition 3.5. (i) Let {(Xi,Ti): i ∈ I} be a family of topological spaces with Tychonoff product∏X∩. Assume∩ that F is an E-ultrafilter, where C(X) ⊂ E ⊆ K(X). Then ( Fi) ⊆ F, where Fi is given by (1.3). ∏ ∩ i∈I Hence, if ( Fi) ≠ ∅, then F is not free. In particular, if each Xi i∈I is compact and T1, then X has no free C(X)-ultrafilters. Hence, TCT if and only if “For every non-compact T1 space (X,T ) and for every T1 base C of X, there is a free C-ultrafilter F.” (ii) Every non-compact T1 topological space has a free closed ultrafilter implies BPI. ∏ ∩ ∩ Proof. (i) Let x = (xi)i∈I ∈ ( Fi). We show that x ∈ F. To this i∈I end, fix F ∈ F. Since C(X) is a base for the closed sets of X, we express F as

F = ∩{Zj : j ∈ J}, ∈ ∪{ −1 ∈ } ∈ <ω where for all j J, Zj = πq (Fqj ): q Qj , Qj [I] , and for all ∈ c ∈ F ∈ F q Qj, Fqj Tq. Since is a ﬁlter and F , it follows that for every j ∈ J, Zj ∈ F. WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 285

We shall show that x ∈ Zj for all j ∈ I; hence, x ∈ F . To this end, fix ∈ F E an index j0 J. As Qj0 is finite and is an -ultrafilter, it follows that ∈ −1 ∈ F there is a q∗ Qj0 such that πq∗ (Fq∗j0 ) . Then −1 ∈ F Fq∗j0 = πq∗ (πq∗ (Fq∗j0 )) q∗ . ∩ ∩ F ⊆ ∈ F ∈ Thus, q∗ Fq∗j0 and since x(q∗) q∗ , we have that x(q∗) Fq∗j0 . Hence, x ∈ π−1(F ) ⊂ ∪{π−1(F ): q ∈ Q ,F c ∈ T } = Z . q∗ q∗j0 q qj0 j0 qj0 q j0 ∈ ∈ ∩{ ∈ Thus, x Zj0 and since∩j0 was arbitrary, it follows that x Zj : j J} = F . Therefore, x ∈ F as required. To see the last assertion, fix F a C(X)-ultrafilter. Then, by Proposition 3.4, each Fi is a closed ultrafilter of Xi. For all i ∈ I, let xi be the unique element of ∩Fi. (Recall that Xi is compact. Thus, all closed ultrafilters of Xi are not free and singletons are closed since Xi is a T1 space.)∩ By (i), the element x ∈ X, satisfying for all i ∈ I, x(i) = xi, is in F. Hence, F is not free as required. (ii) If for some X, 2X is not compact, then, by our hypothesis, 2X has a free closed ultrafilter F. As each Fi is clearly a closed ultrafilter (canonical projections of closed subsets of 2X are closed), it follows, similarly to the proof of (i), that F is not free. This is a contradiction showing that 2X is compact.

Proposition 3.6. Let {(Xi,Ti): i ∈ I} be a family of compact topological spaces with Tychonoﬀ product X.

(i) If F is a CR(X)-ultrafilter, then F = F∩C(X) is a C(X)-ultrafilter. (ii) If F, H, and F ̸= H are CR(X)-ultrafilters, then F ≠ H = H ∩ C(X). (iii) If F is a C(X)-ultrafilter, then the filter F generated by the col- <ω lection S = {∩FQ : Q ∈ [I] and Q is a set of restricted coordinates } F { ∈ } C of some member of F , where Q = F F : QF = ∏Q , is a R(X)- ultrafilter. (Note that ∩FQ ≠ ∅ by the compactness of Xi for a finite i∈Q subset Q of I.) Proof. (i) This follows at once from [9, Proposition 3.2(ii)]. (ii) Since F ̸= H, there exist F ∈ F and H ∈ H with F ∩ H = ∅. By Proposition 2.9, express F and H as ∩{∪{ −1 ∈ } ∈ } (3.2) F = πq (Fjq): q QF : j J and ∩{∪{ −1 ∈ } ∈ } H = πq (Fvq): q QH : v V , 286 K. KEREMEDIS AND E. TACHTSIS

∈ ∈ c ∈ ∈ ∈ where for each j J and q QF , Fjq Tq, and for each v V and∏ q c ∈ ∪ QH , Fvq Tq. Let Q = QF QH . By the compactness of YQ = Xq, q∈Q K {∪{ −1 ∈ } ∈ } W there exist finite subsets of πq (Fjq): q QF : j J and of {∪{ −1 ∈ } ∈ } ∩ K∪W ∩K ∪ ∩W ∅ πq (Fvq): q QH : v V such that ( ) = ( ) ( ) = . Otherwise, F ∩ H ≠ ∅, contradicting the hypothesis F ∩ H = ∅. Since ∩ K ∈ F and ∩ W ∈ H, we see that F ≠ H as required. ∏ (iii) We note, by the compactness of YQ = Xq for finite Q, that S q∈Q has the fip. Assume that F ∈ CR(X) meets non-trivially each member of F. We show that F ∈ F. By Proposition 2.9, express F in the form {∪{ −1 of (3.3). Since F is a C(X)-ultrafilter, it follows that πq (Fjq): ∈ } ∈ } ⊂ F ∩{∪{ −1 ∈ q QF : j J QF , and consequently F = πq (Fjq): q } ∈ } ⊇ ∩F ∈ F F C QF : j J QF . Hence, F and is a R(X)-ultrafilter as required.

4. On the Existence of Wallman Compactifications of T1 Spaces and the Axiom of Choice Our first result in this section shows that the existence of Wallman compactifications of T1 spaces is equivalent to AC. Theorem 4.1. In ZF, the following statements are pairwise equivalent. (i) AC. (ii) For every T1 space (X,T ) and for every T1 base C for X, W(X, C) is a compactification of X. (iii) For every T1 space (X,T ) and for every T1 base C of X, every filter base G ⊂ C extends to a C-ultrafilter F. (iv) TCT. (v) For every non-compact T1 space (X,T ) and for every T1 base C of X, there is a free C-ultrafilter F. Proof. The implications (i) → (ii) ↔ (iii) are established in Theorem 2.4. (iii) → (iv) has been established in [9, Theorem 5.1]. (iv) ↔ (i) is Proposition 2.11(i) and (iv). (iv) → (v) is straightforward. Finally, by Proposition 3.5(i), we have (v) ↔ (iv).

Combining Theorem 2.3 and Theorem 4.1, we get the following straightforward corollary. Corollary 4.2. In ZF, the following statements are pairwise equivalent. (i) AC. WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 287

(ii) For every T1 space (X,T ) and every T1 base C for X, W(X, C) is a compactification of X. (iii) “For every T1 space (X,T ), W(X) is a compactification of X” + CAC. (iv) CFE1 + CAC. (v) “Every non-compact T1 topological space has a free closed ultrafilter” + CAC.

In particular, under CAC, the statements CFE1, “Every non-compact T1 topological space has a free closed ultrafilter,” “For every T1 space (X,T ) and every T1 base C for X, W(X, C) is a compactification of X,” and “For every T1 space (X,T ), W(X) is a compactification of X” are pairwise equivalent.

Proof. We only show that (v) implies (i). Fix A = {Ai : i ∈ I} a pairwise disjoint family of nonempty sets. For every i ∈ I, let Xi = Ai ∪ {∗i}, ∗i ∈/ {{∗ } ∅} ∪ { ⊆ | c| ℵ } ∈ Ai and Ti = i , Z Xi : Z <∏ 0 . Clearly, for every i I, (Xi,Ti) is a compact T1 space. Put X = Xi. Follow now the proof of i∈I Proposition 2 in [6] in order to verify that canonical projections of closed subsets of X are closed. (We indicate here that this is the only point where CAC is needed in the proof of (v) → (i).) It follows, by Proposition B { −1 ∈ } 3.5 and our hypothesis, that X is compact. Since = πi (Ai): i ∩I is a family of closed subsets of X with the ﬁp, it follows that any f ∈ B is a choice function of A, ﬁnishing the proof of the corollary.

In the next theorem we show that “Every non-compact T1 topological space has a free closed ultrafilter” implies “ every infinite set has a countably infinite subset” which is known to be a consequence of CAC ; see [7].

Theorem 4.3. “Every non-compact T1 topological space has a free closed ultrafilter”; hence, CFE1, implies every infinite set has a countably infinite subset. Proof. Assume the contrary and let A be an infinite set without countably i infinite subsets. For every i ∈ N, let Yi = Zi ∪{∗}, where Zi = {f ∈ A : f is 1 : 1} and ∗ ∈/ ∪{Zi : i ∈ N}. Clearly, for each i ∈ N, (Yi,Qi), where Q = {{∗}, ∅} ∪ {Z ⊆ Y : |Zc| < ℵ }, i ∏ i 0 is a compact T1 space. Put Y = Yi. If Y is compact, then S = i∈N { −1 ∈ N} ∩πi (Zi): i is a family of closed subsets of Y with the fip; hence, S= ̸ ∅ and any element in the latter intersection easily yield an injection from N into A, which contradicts our assumption. So we may assume that Y is not compact. Furthermore, as A has no countably infinite subsets, 288 K. KEREMEDIS AND E. TACHTSIS

any infinite subfamily of S has empty intersection. By our hypothesis, let V be a free closed ultrafilter of Y . We assert that canonical projections of elements of V are closed sets. Assume not. Then there is an i ∈ N and a V ∈ V such that πi(V ) is infinite and Zi\πi(V ) ≠ ∅. Express V as

V = ∪{Vn : n ∈ N},Vn = {x ∈ V : ∀m ≥ n, x(m) = ∗}

Note that for every n ∈ N, Vn is a closed subset of Y . Indeed, let n ∈ N ∈ c ∈ c c and let x Vn . If x V , then V is a neighborhood of x avoiding Vn. If ∈ \ ≥ ∈ −1 x V Vn, then there is m n such that x(m) Zm. Clearly, πm (Zm) is a neighborhood of x missing Vn. For every n ∈ N, πi(Vn) is a closed subset of Yi. Indeed, ﬁx n ∈ N. If i ≥ n, then πi(Vn) = {∗} which is closed in Yi. Assume that i < n { − } and let S = 1, 2, ..., n 1 . Since Vn is closed in∏ Y, it follows that Vn × × · · · × × {∗} is closed in the subspace Y1 Y2 Yn−1 j≥n ∏of Y which is topologically homeomorphic to the compact space WS = j∈∏S Yj. Thus, the projection pS(Vn) of Vn onto WS is a closed set in W∏S. As j∈S\{i} Yj × is compact, it follows that the projection πi of Yi j∈S\{i} Yj (which is homeomorphic to WS) onto Yi is a closed map (see Proposition 2.10 (ii)). Hence, πi(pS(Vn)) is closed in Yi. Since πi(Vn) = πi(pS(Vn)), we have that πi(Vn) is closed in Yi as asserted. Since for all n ∈ N, Zi ≠ πi(Vn) and πi(Vn) is closed in Yi, it follows that πi(Vn) ∩ Zi is ﬁnite. Therefore,

πi(V ) ∩ Zi = πi(∪{Vn : n ∈ N}) ∩ Zi = ∪{πi(Vn) ∩ Zi : n ∈ N}

is an infinite subset of Zi which is expressible as a countable union of finite sets. Since our hypothesis implies ACfin (see Proposition 3.5 and Theorem 2.4), we have that πi(V ) ∩ Zi is a countably infinite subset of Zi, say πi(V ) ∩ Zi = {fn : n ∈ N}. Clearly, | ∪ {Ran(fn): n ∈ N}| = ℵ0, and consequently A has a countably infinite subset, contradicting our assumption on A. Thus, canonical projections of elements of V are closed sets and we may follow the proof of Corollary 4.2 in order to verify that Y is compact. This contradicts our assumption that Y is not compact. Therefore, A has a countably infinite subset, finishing the proof of the theorem.

Remark 4.4. From Theorem 4.3, we conclude that in ZF, BPI implies neither CFE1 nor “ Every non-compact T1 topological space has a free closed ultrafilter.” Indeed, in the basic Cohen model M1 in [7], BPI is true, whereas the set of the added Cohen reals has no countably infinite subsets. Thus, “ Every non-compact T1 topological space has a free closed ultrafilter”; hence, CFE1 also fails in M1. WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 289

5. Wallman Compactifications of Tychonoff Products In [9, Theorem 5.1] it is shown that the statement “for every fam- { ∈ } H ⊂ ily (Xi,Ti):∏ i I of compact topological spaces, every family C(X),X = Xi, with the fip extends to a maximal family F ⊂ C(X) i∈I { ∈ } with the fip” (or, equivalently, “for every family (Xi,T∏i): i I of compact topological spaces, every C(X)-filter H of X = Xi extends to a i∈I { ∈ } C(X)-ultrafilter”) and the proposition “for every family (X∏i,Ti): i I of compact topological spaces, every CR(X)-filter H of X = Xi extends i∈I C to a R(X)-ultrafilter” are both equivalent to TCT. Hence, for every∏ family {(Xi,Ti): i ∈ I} of compact T1 spaces, W(X, C(X)), X = Xi, ∏ i∈I is compact if and only if Xi is compact if and only if W(X, CR(X)) i∈I is compact. It is straightforward to see that TCT implies that for every family {(X ,T ): i ∈ I} of compact T spaces with product X, i i 1 ∏ ∏ (5.1) W(X, C(X)) ≃ W(X, CR(X)) ≃ W(Xi) ≃ Xi. i∈I i∈I We∏ show in the forthcoming Theorem 5.2 that for a Tychonoff product Xi of compact T1 spaces, (5.1) is actually a ZF-result. i∈I { Our first result in this section shows∏ that∏ “For every family (Xi,Ti): i ∈ ω} of compact T1 spaces, W( Xi) ≃ W(Xi)” is a consequence i∈ω i∈ω of the countable axiom of choice CAC. Theorem 5.1. (i) For any index set I, AC∏(I) implies∏ “For every family {(Xi,Ti): i ∈ I} of compact T1 spaces, W( Xi) ≃ W(Xi).” i∈I i∈I ∏ (ii) “For every family {(Xi,Ti): i ∈ ω} of compact T1 spaces, W( Xi) ∏ i∈ω ≃ W(Xi)” does not imply “For every T1 space (X,T ), W(X) is a com- i∈ω pactification of X” in ZF. (iii) CAC + “In a countable product of compact T1 spaces, every filter of closed sets extends to a closed ultrafilter” if and only TCT restricted to countable families of compact topological spaces. W (iv) “For every T1 space (X,T ), (X) is a compactification∏ of X” + “For every family {(Xi,Ti): i ∈ ω} of compact T1 spaces, W( Xi) ≃ ∏ i∈ω W(Xi)” can be added to the list of Corollary 4.2. i∈ω { ∈ } Proof. (i) Fix a set I and let∏ (Xi,Ti): i I be a family of compact T1 spaces with product X = Xi. Let F be a closed ultrafilter of X. For i∈I 290 K. KEREMEDIS AND E. TACHTSIS ∩ every i ∈ I, let Ai = Fi where Fi is given by (1.3). Since Xi is compact, ̸ ∅ A { ∈ } it follows that Ai = . Put = Ai : i I and,∩ by AC(I), let x be a choice function of A. By Proposition 3.5, x ∈ F, and consequently F is not free. Hence, X has no free closed ultrafilters and, by Proposition 3.1, W(X) ≃ X. Since for every i ∈ I,Xi is compact, the function φi : Xi → W(Xi), F F φi(x) =∏ (x) where∏ (X) is given by (2.6), is a homeomorphism.∏ ∏ Thus, X = Xi ≃ W(Xi) under the mapping ϕ : Xi → W(Xi), i∈ω i∈ω ∏ ∏i∈ω i∈ω ϕ(x)(i) = φi(x(i)). Hence, W(X) ≃ X = Xi ≃ W(Xi) as required. i∈I i∈I (ii) It is known that in Model M47(n,M) in [7], there exists a family {(Xi,Ti): i ∈ I} of compact T1 spaces whose product X is not compact (it is known that BPI, and hence by Theorem 2.4 “ For every T1 space (X,T ), W(X) is a compactification of X” also, fails in M47(n,M)) but { ∈ } CAC holds. However, by∏ (i), “ For∏ every family (Xi,Ti): i ω of compact T1 spaces, W( Xi) ≃ W(Xi)” holds in M47(n,M). i∈ω i∈ω (iii) It suffices to show (→ ) as the other implication is straightforward. Let {(Xi,Ti): i ∈ ω} and X be as in (i). Let G be a family of closed subsets of X with the fip. Let F be a closed ultrafilter which includes G. As in the proof of (i), we may show that ∩F ̸= ∅; hence, ∩G ̸= ∅ and X is compact as required. (iv) First note that our hypothesis implies TCT∏ restricted∏ to countable families of compact topological spaces (X = Xi ≃ W(Xi); hence, i∈ω i∈ω by our hypothesis, W(X) ≃ X and by “For every T1 space (X,T ), W(X) is a compactification of X,” W(X), hence X, is compact). The conclusion now follows from Theorem 2.4(iii) and Corollary 4.2.

The next theorem indicates a link between Wallman compactifications and Tychonoff’s compactness theorem. { ∈ } Theorem 5.2. (i) (ZF) For every family∏ (Xi,Ti): i∏ I of compact T1 spaces with product X, W(X, C(X)) ≃ W(Xi) ≃ Xi. In particular, i∈I i∈I “For every T1 space (X,T ) and every T1 base C for X, W(X, C) is a compactification of X” if and only if TCT. (ii) (ZF) For every family {(Xi,Ti): i ∈ I} of compact T1 spaces with product X, W(X, CR(X)) ≃ W(X, C(X)). { ∈ } (iii) Assume that for every family (Xi,Ti): i I of∏ compact T1 topological spaces, for every closed ultrafilter F of X = Xi, and for i∈I every i ∈ I, Fi, given by (1.3), is a closed ultrafilter of Xi. Then W(X) ≃ WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 291 ∏ ∏ W(Xi) ≃ Xi. In addition, if we add to our hypothesis, “Every non- i∈I i∈I compact T1 topological space has a free closed ultrafilter,” then TCT holds. (iv) AC if and only if “Every non-compact T1 topological space has a free { ∈ } closed ultrafilter” + “for every family (Xi,Ti): i I of infinite∏ compact T1 topological spaces, for every closed ultrafilter F of X = Xi, and i∈I for every i ∈ I, |Fi| > 2, where Fi is given by (1.3).” { ∈ } Proof.∏ (i) Fix (Xi,Ti): i I a family of compact T1 spaces∏ and let X = Xi be their product. As in the proof of Theorem 5.1(i), W(Xi) ≃ i∏∈I i∈I Xi. Let F be a C(X)-ultrafilter of X. By Proposition 3.4, it follows i∈I that for all i ∈ I, the collection Fi given by (1.3) is a closed ultrafilter and since Xi is a compact T1 space, ∩Fi is a singleton, say {xi}, i ∈ I. ∈ ∩F By Proposition 3.5, (xi)i∈I . Thus, ∏X has no free C(X)-ultrafilters, and by Proposition 3.1, W(X, C(X)) ≃ Xi. i∈I (ii) Let h : W(X, CR(X)) → W(X, C(X)) be the mapping given by h(F) = F = F ∩ C(X). By (i), (ii), and (iii) of Proposition 3.6, h is well defined, 1 : 1, and onto. We show now that h is a closed mapping. Let <ω A ∈ CR(X). Then by (i) of Proposition 2.9, there is a Q ∈ [I] such ∩{ ∈ } ∈ ∪{ −1 c ∈ that A = Uj : j J , where for all j J, Uj = πq (Fjq): Fjq ∗ Tq, q ∈ Q}. It follows that h(A ) = h({F ∈ W(X, CR(X)) : A ∈ F}) = ∩{F ∈ W ∈ F} ∩{ ∗ ∈ } ∗ (X, C(X)) : Uj = Uj : j J . Thus, h(A ) is closed in W(X, C(X)), being an intersection of closed subsets of W(X, C(X)), and consequently h is closed as required. Similarly, one may show that h is continuous. Hence, h is a homeomorphism. (iii) As in part (i), our hypothesis implies that X∏has no free∏ closed ultrafilters. Hence, by Proposition 3.1, W(X) ≃ W(Xi) ≃ Xi. i∈I i∈I Since X has no free closed ultrafilters, our second hypothesis implies that X is compact. (iv) It suffices to show (←) as the other implication is straightforward. To this end, it suffices, in view of Corollary 4.2(v), to show that CAC holds. Fix A = {Ai : i ∈ ω} a family of pairwise disjoint nonempty sets. for every i ∈ ω, let Xi, Ti, X, and B be as in the proof of Corollary 4.2. If X is compact, then any f ∈ ∩B is a choice function of A. If X is not compact, then X has a free closed ultrafilter F. For∩ every i ∈ ω, let Fi be given by (1.3). Clearly, for every i ∈ ω, Vi = Fi ≠ ∅. If ∞ |B ∩ F| < ℵ0, then for all i ∈ ω, Vi = {∗i} (Xi = Ai ∪ {∗i} and by the F −1 ∈ F −1 {∗ } ∈ F ∈ maximality of , either πi (Ai) or πi ( i ) ). For every i ω with Vi ≠ {∗i}, fix ai ∈ Vi and let a ∈ X satisfy a(i) = ai if Vi ≠ {∗i} 292 K. KEREMEDIS AND E. TACHTSIS

and a(i) = ∗i otherwise. As in Proposition 3.5, it follows that a ∈ ∩F. Hence, F is not free. This is a contradiction showing that |B∩F| = ℵ0. In view of the equivalence between PCAC and CAC, we may assume, for our convenience, that B ⊂ F. Since |Fi| > 2 for all i ∈ ω, it follows easily that each Vi is a finite nonempty subset of Ai. Hence, by “Every non-compact T1 topological space has a free closed ultrafilter,” Proposition 3.5(ii), and Theorem 2.4(iv), {Vi : i ∈ ω} has a choice function f. Clearly, f is also a choice function of A, finishing the proof of (iv) and the proof of the theorem.

The question which arises at this point is whether the compactness requirement in Theorem 5.2(i) can be dropped. We show next that this can be done.

{ ∈ } Theorem 5.3 (ZF). For every family (∏Xi,Ti): i I of T1 spaces, their product X satisfies W(X, C(X)) ≃ W(Xi). In particular, TCT i∈I ∏ implies “for every family {(Xi,Ti): i ∈ I} of T1 spaces, W(Xi) is a ∏ i∈I Wallman compactification of the product Xi,” a special case of a more i∈I general ZFC result established in [11]. ∏ Proof. We show that the mapping h : W(X, C(X)) → W(Xi) (given i∈I by h(F) = (Fi)i∈I , where for all i ∈ I, Fi is given by (1.3)) is a homeomorphism. By Proposition 3.4, h is well defined, 1 : 1, and onto. −1 ∗ c ∈ Additionally,∏ h is continuous. Fix πi (A ),A∏ Ti a subbasic closed W −1 ∗ { F ∈ W ∈ F } set of (Xi). Since πi (A ) = ( q)q∈I (Xq): A i , we i∈I q∈I −1 −1 ∗ {F ∈ W −1 ∈ F} −1 ∗ see that h (πi (A )) = (X, C(X)) : πi (A) = (πi (A)) is a closed subset of W(X, C(X)). To complete the proof of the theorem, it suffices to show that h is −1 ∗ c ∈ W closed. Fix (πi (A)) and A Ti a subbasic closed set of (X, C(X)). −1 ∗ {F ∈ W −1 ∈ F} Since (πi (A)) = ∏(X, C(X)) : πi (A) , we see that −1 ∗ { F ∈ W ∈ F } −1 ∗ h((πi (A)) ) = ( q)q∈I (Xq): A i = πi (A ) is a closed ∏ q∈I subset of W(Xi) as required. i∈I

Question 5.4. (i) Do “ For every T1 space (X,T )” and “W(X) is a com- { ∈ } pactiﬁcation of∏X” imply∏ “For every family (Xi,Ti): i ω of compact T1 spaces, W( Xi) ≃ W(Xi)”? i∈ω i∈ω WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 293

{ ∈ } (ii)∏ Do “ For∏ every family (Xi,Ti): i ω of compact T1 spaces” and “W( Xi) ≃ W(Xi)” imply CAC ? i∈ω i∈ω { ∈ } (iii)∏ Is “ For∏ every family (Xi,Ti): i ω of compact T1 spaces, W( Xi) ≃ W(Xi)” provable in ZF ? i∈ω i∈ω (iv) Does CAC imply the statement, “ In a countable product of compact T1 spaces, every ﬁlter of closed sets extends to a closed ultraﬁlter” ?

6. Countably Closed and Countably Prime Closed Ultrafilters in T1 Spaces The proof of Theorem 4.3 indicates that if the closed ultrafilter V is countably prime, then there exists V ∈ V which has finite projections on some coordinate spaces. Hence, the existence of countably prime closed ultrafilters in products of compact T1 spaces might lead to compactness of their product. We prove in this section that this is the case. Proposition 6.1. (i) Let (X,T ) be a compact topological space. Then every closed ultrafilter F of X is countably closed and countably prime. (ii) Assume that for every family A = {Ai : i ∈ ω} of nonempty sets, there exists a family B = {Bi : i ∈ ω} of countable nonempty sets such that Bi ⊆ Ai for all i ∈ ω. Let (X,T ) be a topological space and let F be a closed ultrafilter of X which is countably closed. Then F is countably prime. ∩ Proof. (i) Let F be a closed ultrafilter of X. Since X is compact,∩ F is a nonempty closed subset of X and, as F is a closed ultrafilter, F ∈ F. It follows that F is countably closed. To see that F is countably prime, fix {Gi : i ∈ ω} a countable family of nonempty closed subsets of X such that G = ∪{Gi : i ∈ ω} ∈ F. We show that Gi ∈ F for some i ∈ ω. Assume the contrary. Then, for all i ∈ ω, {F ∈ F : F ∩ Gi = ∅} ̸= ∅. For every i ∈ ω, put

Fi = ∩{F ∈ F : F ∩ Gi = ∅}. ∩ ⊇ F ∈ F ∈ ∈ F Since∩Fi , it follows that for every i ω, Fi . Hence, F = {Fi : i ∈ ω} ∈ F and F ∩ G ∈ F. But

F ∩ G = F ∩ (∪{Gi : i ∈ ω}) =

∪{F ∩ Gi : i ∈ ω} ⊆ ∪{Fi ∩ Gi : i ∈ ω} = ∅ and F ∩ G/∈ F. This is a contradiction ﬁnishing the proof of (i).

(ii) Fix {Gi : i ∈ ω} a family of nonempty closed subsets of X satisfying G = ∪{Gi : i ∈ ω} ∈ F. We show that there exists i ∈ ω with Gi ∈ F. Assume on the contrary that for every i ∈ ω, Gi ∈/ F. It follows, by the 294 K. KEREMEDIS AND E. TACHTSIS

maximality of F, that for every i ∈ ω, ∅ ≠ Ai = {F ∈ F : Gi ∩ F = ∅}. By our hypothesis and the fact that F is countably closed, it is easy to see that there exists a choice function f of the family {Ai : i ∈ ω}. Put F = ∩{f(i): i ∈ ω}. Since F is countably closed, it follows that F ∈ F, and consequently F ∩ G ≠ ∅. Similarly to the proof of (i), we may show that F ∩ G = ∅, a contradiction. This completes the proof of (ii) and the proof of the proposition. Remark 6.2. Clearly, in ZFC, T = {(x, +∞): x ∈ R} ∪ {∅, R} is a topology on the real line R and F = {(−∞, x]: x ∈ R} ∪ {R} is a countably prime closed ultrafilter of (R,T ) which fails to be countably closed. Thus, in ZFC, F is countably prime 9 F is countably closed. Theorem 6.3. In ZF, the following are equivalent. (i) AC. { ∈ } (ii) For every family (Xi,Ti): i I ∏of compact T1 spaces, every closed filter H of the product X = Xi extends to a closed ul- i∈I trafilter F which is countably closed and countably prime. { ∈ } (iii) For every family (Xi,Ti): i I ∏of compact T1 spaces, every closed filter H of the product X = Xi extends to a closed ul- i∈I trafilter F which is countably closed. ∏ Proof. (i) → (ii). Let X = Xi be the product of the compact T1 spaces i∈I Xi. By AC, X is compact and every closed filter is extended to a closed ultrafilter. The conclusion follows from Proposition 6.1. (ii) → (iii) is straightforward.

(iii) → (i). We show first that CAC holds. Fix A = {Ai : i ∈ ω} a family of nonempty sets. For every i ∈ ω, let Xi = Ai ∪ {∗i}, ∗i ∈/ Ai, {{∗ } ∅} ∪ { ⊆ | c| ℵ } ∈ and Ti = i , Z Xi : Z < ∏0 . Clearly, for every i ω, (Xi,Ti) is a compact T1 space. Put X = Xi and let H be the closed i∈ω S { −1 ∈ } F filter generated by the family = πi (Ai): i ω . Let be a closed, countably closed ultrafilter extending H. Since S ⊂ F, it follows that ∩S ∈ F. Hence, ∩S= ̸ ∅, and consequently A has a choice function as required. We show now that AC holds. Fix A = {Ai : i ∈ I} a family of nonempty sets. For every i ∈ I, let Xi, Ti, X, S, and F be as in the first part of this proof. In view of the proof of Proposition 2 in [6], it suffices, to show that canonical projections of closed sets are closed. Since CAC holds, the proof of Proposition 2 in [6] shows that we further need to establish that there is a free ultrafilter on ω. However, following the proof of Proposition 2.13, we have that (iii) of the present theorem implies BPI, WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 295

which in turn implies that there is a free ultraﬁlter on ω. Now, as in the proof of Proposition 2 of [6], we may deﬁne a choice function of A. This completes the proof of the theorem.

Corollary 6.4. The following are equivalent. (i) AC. { ∈ } (ii) For every family (Xi,Ti): i I of compact∏T1 spaces, every filter H of closed subsets of the product X = Xi extends to a i∈ω closed ultrafilter F which is countably closed and countably prime. { ∈ } (iii) For every family (Xi,Ti): i I of compact∏T1 spaces, every filter H of closed subsets of the product X = Xi extends to a i∈ω closed ultrafilter F which is countably prime. Proof. Since (i) and (ii) of the corollary coincide with (i) and (ii), respec- tively, of Theorem 6.3, it suffices to show that (iii) → (i). Furthermore, since (iii) implies BPI (as in the proof of Theorem 6.3), we only show that CAC holds. To see this, fix A = {Ai : i ∈ ω} a disjoint family of infinite sets. Assume that A has no partial choice sets. For every i ∈ ω, let Xi, Ti, and X be as in the proof of Theorem 6.3 and let F be a countably prime closed ultrafilter of X which extends the closed filter H generated S { −1 ∈ } by the family = πi (Ai): i ω . Claim. ∈ ∈ −1 { } ∈ For every i ω, there is a unique ai Ai such that πi ( ai ) F.

Proof of the claim. Fix i ∈ ω and let Gi = {F ∈ F : Ai\πi(F ) ≠ ∅}. ̸ ∅ ∅ ∈ F ∪ { −1 { } } Clearly, Gi = . (If Gi = , then for every a Ai, πi ( a ) has F −1 { } ∈ F the ﬁp, and consequently, by the maximality of , πi ( a ) , which is a contradiction.) Fix F ∈ Gi and express F as F = ∪{Fn : n ∈ ω},

(6.1) Fn = {f ∈ F : ∀m ≥ n, f(m) = ∗m}.

Since each Fn is closed (as in the proof of Theorem 4.3), F ∈ F, and F is countably prime, it follows that for some n ∈ ω, Fn ∈ F. Since Ki = πi(Fn)∩Ai is a nonempty closed proper subset of Ai, it follows that −1 ∈ F Ki is a ﬁnite subset of Ai and πi (Ki) . Thus, by the maximality of F ∈ −1 { } ∈ F , there exists a unique ai Ai with πi ( ai ) , ﬁnishing the proof of the claim.

For every i ∈ ω, let ai be the unique element of Ai which is guaranteed by the claim. Clearly, {ai : i ∈ ω} is a choice set of A, contradicting our hypothesis on A having no partial choice sets. Therefore, CAC holds and the proof of the corollary is complete. 296 K. KEREMEDIS AND E. TACHTSIS

7. Open Questions and Directions for Further Study Below, we summarize the open problems mentioned throughout the paper.

(1) Does “ For every T1 space (X,T ), W(X) is a compactification of X” imply AC ? Equivalently (see Theorem 2.4), does CFE1 imply AC ? Equivalently (see Theorem 4.1), does “For every T1 space (X,T ), W(X) is a compactification of X” imply “For every T1 space (X,T ) and every T1 base C for X, W(X, C) is a compactification of X”? (2) Does “ For every T1 space (X,T ), W(X) is a compactification { ∈ } of X” imply∏ “For every∏ family (Xi,Ti): i ω of compact T1 spaces, W( Xi) ≃ W(Xi)”? i∈ω i∈ω { ∈ } (3) Does∏ “ For every∏ family (Xi,Ti): i ω of compact T1 spaces, W( Xi) ≃ W(Xi)” imply CAC ? i∈ω i∈ω { ∈ } (4) Is “∏ For every∏ family (Xi,Ti): i ω of compact T1 spaces, W( Xi) ≃ W(Xi)” provable in ZF ? i∈ω i∈ω (5) Does CAC imply the statement, “In a countable product of compact T1 spaces, every filter of closed sets extends to a closed ultrafilter” ?

For Wallman-type compactiﬁcations and their relationship to AC or to certain weak forms of AC, the reader is referred to the following related papers: [1], [2], [4], [5].

Acknowledgment. We are grateful to the anonymous referee for several useful suggestions which improved the exposition of our paper.

References

[1] Eric J. Hall and Kyriakos Keremedis, Čech-Stone compactifications of discrete spaces in ZF and some weak forms of the Boolean prime ideal theorem, Topology Proc. 41 (2013), 111–122. [2] Eric J. Hall, Kyriakos Keremedis, and Eleftherios Tachtsis, The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters. To appear in Mathematical Logic Quarterly. [3] Horst Herrlich, The axiom of choice holds iff maximal closed filters exist, MLQ Math. Log. Q. 49 (2003), no. 3, 323–324.

[4] Horst Herrlich and Kyriakos Keremedis, Remarks on the space ℵ1 in ZF, Topology Appl. 158 (2011), no. 2, 229–237. WALLMAN COMPACTIFICATIONS AND TYCHONOFF PRODUCTS IN ZF 297

[5] Horst Herrlich, Kyriakos Keremedis, and Eleftherios Tachtsis, Remarks on the Stone spaces of the integers and the reals without AC, Bull. Pol. Acad. Sci. Math. 59 (2011), no. 2, 101–114. [6] Horst Herrlich and Juris Stepr¯ans, Maximal filters, continuity and choice princi- ples, Quaestiones Math. 20 (1997), no. 4, 697–705. [7] Paul Howard and Jean E. Rubin, Consequences of the Axiom of Choice. Math- ematical Surveys and Monographs, 59. Providence, RI: American Mathematical Society, 1998. [8] J. L. Kelley, The Tychonoff product theorem implies the axiom of choice, Fund. Math. 37 (1950), 75–76. [9] Kyriakos Keremedis, Tychonoff products of compact spaces in ZF and closed ultrafilters, MLQ Math. Log. Q. 56 (2010), no. 5, 475–487. [10] Kyriakos Keremedis and Eleftherios Tachtsis, On the extensibility of closed filters in T1 spaces and the existence of well orderable filter bases, Comment. Math. Univ. Carolin. 40 (1999), no. 2, 343–353. [11] Frank Kost, Wallman-type compactifications and products, Proc. Amer. Math. Soc. 29 (1971), 607–612. [12] James R. Munkres, Topology: A First Course. Englewood Cliffs, N.J.: Prentice- Hall, Inc., 1975. [13] Jan Mycielski, Two remarks on Tychonoff’s product theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 439–441. [14] Jun-iti Nagata, Modern General Topology. 2nd ed. North-Holland Mathematical Library, 33. Amsterdam: North-Holland Publishing Co., 1985. [15] Herman Rubin and Dana Scott, 558. Some topological theorems equivalent to the Boolean prime ideal theorem, preliminary report, in “May Meeting in Yosemite” (386–399), Bull. Amer. Math. Soc. 60 (1954), no. 4, 389. [16] Eric Schechter, Kelley’s specialization of Tychonoff’s Theorem is equivalent to the Boolean Prime Ideal Theorem, Fund. Math. 189 (2006), no. 3, 285–288. [17] Henry Wallman, Lattices and topological spaces, Ann. of Math. (2) 39 (1938), no. 1, 112–126. [18] Stephen Willard, General Topology. Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581]. Mineola, NY: Dover Publications, Inc., 2004.

(Keremedis) University of the Aegean; Department of Mathematics; Karlovassi, Samos 83200, Greece E-mail address: [email protected]

(Tachtsis) University of the Aegean; Department of Statistics and Actuarial- Financial Mathematics; Karlovassi, Samos 83200, Greece E-mail address: [email protected]