On extensions of countable filterbases to ultrafilters and ultrafilter compactness. Issue 2 (19th February 2018)

• Record Type: Journal Article
• Authors: Herrlich, Horst; Howard, Paul; Keremedis, Kyriakos
• Is Part Of: Quaestiones mathematicae. Volume 41:Issue 2(2018)
• Page Start: 213
• Page End: 225
• Publication Date: 2018-02-19
• Subjects: 03E25 -- 03E35 -- 06E15 -- 54D30; Boolean prime ideal theorem -- Stone space -- compactness -- ultrafilter compactness -- Tychonoff compactness theorem -- weak forms of the axiom of choice; Mathematics -- Periodicals; 510.5
• DOI: 10.2989/16073606.2017.1376229
• Languages: English
• ISSNs: 1607-3606

Abstract

Abstract: We show in the Zermelo-Fraenkel set theoryZF without the axiom of choice: Given an infinite set X, the Stone space S ( X ) is ultrafilter compact. For every infinite set X, every countable filterbase of X extends to an ultra-filter iff for every infinite set X, S ( X ) is countably compact. ω has a free ultrafilter iff every countable, ultrafilter compact space is countably compact. We also show the following: There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S ( X ) is not compact but S ( X ) is countably compact and every countable filterbase of X extends to an ultrafilter. It is relatively consistent withZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ℝ which does not extend to an ultrafilter. Hence, it is relatively consistent withZF that ℝ has free ultrafilters but there exists a countable filterbase of ℝ which does not extend to an ultrafilter.