Tychonoff products of compact spaces in ZF and closed ultrafilters

Abstract

Let {(Xi, Ti): i ∈I } be a family of compact spaces and let X be their Tychonoff product. 𝒞(X) denotes the family of all basic non-trivial closed subsets of X and 𝒞R(X) denotes the family of all closed subsets H = V × ΠXi of X, where V is a non-trivial closed subset of ΠXi and QH is a finite non-empty subset of I. We show: (i) Every filterbase ℋ︁ ⊂ 𝒞R(X) extends to a 𝒞R(X)-ultrafilter ℱ if and only if every family H ⊂ 𝒞(X) with the finite intersection property (fip for abbreviation) extends to a maximal 𝒞(X) family F with the fip. (ii) The proposition “if every filterbase ℋ︁ ⊂ 𝒞R(X) extends to a 𝒞R(X)-ultrafilter ℱ, then X is compact” is not provable in ZF. (iii) The statement “for every family {(Xi, Ti): i ∈ I } of compact spaces, every filterbase ℋ︁ ⊂ 𝒞R(Y), Y = Πi ∈IYi, extends to a 𝒞R(Y)-ultrafilter ℱ” is equivalent to Tychonoff's compactness theorem. (iv) The statement “for every family {(Xi, Ti): i ∈ ω } of compact spaces, every countable filterbase ℋ︁ ⊂ 𝒞R(X), X = Πi ∈ωXi, extends to a 𝒞R(X)-ultrafilter ℱ” is equivalent to Tychonoff's compactness theorem restricted to countable families. (v) The countable Axiom of Choice is equivalent to the proposition “for every family {(Xi, Ti): i ∈ ω } of compact topological spaces, every countable family ℋ︁ ⊂ 𝒞(X) with the fip extends to a maximal 𝒞(X) family ℱ with the fip” (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)