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 fam- ily of all closed subsets of X) is a compactification of X” and f 2 g “ForQ every familyQ (Xi;Ti): i ! of compact T1 spaces, W( Xi) and W(Xi) are topologically homeomorphic.” i2! i2! We also show that “ For every T1 space (X; T ), W(X) is a com- pactification 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 f 2 g QLet (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 i2I closed in case [f −1 2 g 2 <! 2 c 2 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 isQ called restricted closed Qif there is a finite set Q ⊆ I such that F = V × Xi, with V ⊂ YQ = Xi a closed set. Note that for a given i2Qc i2Q 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, j n j ≥ 2 n and (for our convenience)Q suppose that I Q 2 and let i0 I Q. 0 × 0 [ f g 0 × Then F = V Xi, where Q = Q i0 and V = V Xi0 ). i2(Q0)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]<! 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 definedQ so that there is a closed set VF × \f 2 g in YQF such that F = VF Xi, VF = Zj : j J , where c i2(QF ) [f −1 2 g ⊂ 2 2 Zj = πq (Gqj): q QF YQF and for all q QF , for all j J, c 2 2 2 Gqj Tq, and for each q QF , there exists an index j QJ such that <! Gqj =6 ?;Xq. Then any Q 2 [I] for which F = V × Xi with i2(Q)c V closed in YQ is such that Q ⊇ QF ; recall that V is expressible as an 0 [f −1 0 2 g 2 0 intersection of basic closed subsets Zj = πq (Gqj): q Q , j J , of 2 2 0 0 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 \f[f −1 2 g 2 g (1.1) G = πq (Fqi): q Qi : i n 2 N 2 2 <! 2 c 2 where n and for all i n, Qi [I] , and for all q Qi, Fqi Tq, or, equivalently, Y [f\f 2 g 2 f −1 2 gg (1.2) G = f(i): i n : f πq (Fqi): q Qi , i2n 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 Q For every S ⊂ I, pS : X ! YS = Xi will denote the projection of i2S X onto YS. In particular, if S = fig for some i 2 I, then we will denote the projection pS by πi. C1(X) denotes the collection of all 1-basic closed C f −1 2 c 2 g 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 familyT G of closed subsets of X having the finite intersection property (fip) G 6= ?. 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 n f?g is an E-filter if and only if (i) if F1, F2 2 F, then F1 \ F2 2 F. (ii) if F 2 F, F 0 2 E and F ⊆ F 0, then F 0 2 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 n f?g is called an E-filter base if, for 2 H 2 H ⊆ \ every H1;H2 , there is an H3 T such that H3 H1 H2. A filter base H ⊆ E n f?g 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 fFi : i 2 !g ⊆ F, then \fFi : i 2 !g 2 F. An E-filter F is called countably prime if it satisfies the condition: If fFi : i 2 !g ⊂ E and [fFi : i 2 !g 2 F, then there exists i 2 ! such that Fi 2 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) ? 2 B. (ii) If B1;B2 2 B, then B1 \ B2 2 B and B1 [ B2 2 B. (iii) If x2 = B 2 B, then there is Bx 2 B such that x 2 Bx and Bx \ B = ?. f 2 g C If (Xi;Ti): i I is a family of T1 topologicalQ spaces, then R(X) and C(X) are T1 bases for the product X = Xi. Note that C(X) is not a i2I T1 base forQX. If X = Xi is a product of topological spaces and F is an E-ultrafilter, i2I where C(X) ⊂ E ⊆ K(X) (= the family of all closed subsets of X), then for every i 2 I; Fi denotes the family of all closed subsets of Xi whose inverse image under πi is a member of F, i.e., F f ⊆ c 2 −1 2 Fg (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 fF ⊂ C : F is a C-ultrafilterg. 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 = fA∗ : A 2 Cg;A∗ = fF 2 W(X; C): A 2 Fg; is called the Wallman topology corresponding to the base C. Since for every A 2 C, (A∗)c = fF 2 W(X; C): 9F 2 F such that F \ A = ?g, it follows that (1.4) U = fU ∗ : U c 2 Cg and U ∗ = fF 2 W(X; C): 9 F 2 F such that F ⊆ Ug is a base for W(X; C).