Tychonoff products of compact spaces in ZF and closed ultrafilters |
| |
Authors: | Kyriakos Keremedis |
| |
Institution: | University of the Aegean, Department of Mathematics, Karlovassi, Samos 83200, Greece |
| |
Abstract: | Let {(Xi, Ti): i ∈I } be a family of compact spaces and let X be their Tychonoff product. ??(X) denotes the family of all basic non‐trivial closed subsets of X and ??R(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 ?? ? ??R(X) extends to a ??R(X)‐ultrafilter ? if and only if every family H ? ??(X) with the finite intersection property (fip for abbreviation) extends to a maximal ??(X) family F with the fip. (ii) The proposition “if every filterbase ?? ? ??R(X) extends to a ??R(X)‐ultrafilter ?, then X is compact” is not provable in ZF. (iii) The statement “for every family {(Xi, Ti): i ∈ I } of compact spaces, every filterbase ?? ? ??R(Y), Y = Πi ∈IYi, extends to a ??R(Y)‐ultrafilter ?” is equivalent to Tychonoff's compactness theorem. (iv) The statement “for every family {(Xi, Ti): i ∈ ω } of compact spaces, every countable filterbase ?? ? ??R(X), X = Πi ∈ωXi, extends to a ??R(X)‐ultrafilter ?” 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 ?? ? ??(X) with the fip extends to a maximal ??(X) family ? with the fip” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |
| |
Keywords: | Axiom of choice weak axioms of choice compactness countable compactness Tychonoff compactness theorem filter ultrafilter |
|
|