On ultracompact spaces in ZF

Abstract

We work in set theory without the Axiom of Choice (AC) and establish the following results:1.“Products of ultracompact spaces are ultracompact” + “there exists a free ultrafilter on ω” (UF(ω)) is equivalent to AC in ZFA;2.“Products of ultracompact spaces are ultracompact” does not imply AC in ZF;3.AC is equivalent to each of “products of ultracompact spaces are countably compact”, “products of ultracompact spaces are sequentially limit point compact”, “products of ℵ0-bounded spaces are countably compact”, “products of ℵ0-bounded spaces are sequentially limit point compact”, and “products of ℵ0-bounded spaces are ℵ0-bounded” in ZFA;4.UF(ω) is equivalent to each of “every ultracompact space is sequentially limit point compact” and “every ultracompact T2 space is sequentially limit point compact”;5.The Axiom of Countable Choice (ACω) is equivalent to “every ℵ0-bounded space is countably compact”;6.ACω+UF(ω) is equivalent to each of “every ultracompact space is countably compact”, “products of countably many ultracompact spaces are countably compact”, “products of countably many ultracompact spaces are sequentially limit point compact”, and “products of countably many ℵ0-bounded spaces are ℵ0-bounded”;7.Each of “a T3 space is ultracompact, if and only if, it is ℵ0-bounded” and “a Tychonoff space is ultracompact, if and only if, it is ℵ0-bounded” is equivalent to “every filter on ω can be extended to an ultrafilter on ω” (BPI(ω)), and thus each of the above statements is equivalent to the statement “the Stone space βω of all ultrafilters on ω is compact”. Thus we provide the exact characterizations of the above topological results, the former by Vaughan in “Countably compact and sequentially compact spaces”, and the latter by Bernstein in “A new kind of compactness for topological spaces”, as a specific renowned weak choice principle (which is strictly weaker than the Boolean Prime Ideal Theorem in ZF);8.ACω implies Ginsburg's theorem in “Some results on the countable compactness and pseudocompactness of hyperspaces”, namely “a T2 space X is ultracompact, if and only if, the hyperspace CL(X) (the set of non-empty closed subsets of X with the Vietoris topology) is ultracompact”, which in conjunction with UF(ω) implies the weak choice principle ACfinω (AC restricted to countable families of non-empty finite sets).