On extensions of countable filterbases to ultrafilters and ultrafilter compactness

Abstract

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

(i) Given an finnite set X, the Stone space S(X) is ultrafilter compact.
(ii) For every finnite set X, every countable filterbase of X extends to an ultra-filter iff for every finnite set X, S(X) is countably compact.
(iii) ω has a free ultrafilter iff every countable, ultrafilter compact space is countably compact.
We also show the following:
(iv) 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.
(v) It is relatively consistent with ZF 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 with ZF that R has free ultrafilters but there exists a countable filterbase of R which does not extend to an ultrafilter.

Keywords: Boolean prime ideal theorem, Stone space, compactness, ultrafilter compactness, Tychonoff compactness theorem, weak forms of the axiom of choice