The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

Abstract

Let X be an infinite set and let BPI(X) and UF(X) denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by S(X) the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, sans-serifUF(ℵ) iff sans-serifUF(2ℵ). UF(ω) iff “2ω is a continuous image of S(ω)” iff “S(ω) has a free open ultrafilter ” iff “every countably infinite subset of S(ω) has a limit point”. BPI(ω) implies “every open filter on 2ω extends to an open ultrafilter” implies “2ωhas an open ultrafilter” implies UF(ω). It is relatively consistent with sans-serifZF that sans-serifUF(ω) holds, whereas sans-serifBPI(ω) fails. In particular, none of the statements given in (2) implies sans-serifBPI(ω).

On preimages of ultrafilters in ZF

“…Regarding implications, non-implications and equivalent forms of the principles BPI(ω) and UF(ω) we refer the interested reader to [6], [3] and [1]. All principles involving ultrafilters in their definition are easily seen to be consequences of BPI.…”

“…UF(ω) BPI(ω) has been established in [3]. For the non-implications we refer the reader to [6] where a symmetric model N has been constructed in which |R| = ℵ 1 but the set C of all co-countable subsets of ω 1 is included in no ultrafilter of ω 1 meaning that UUF(ω 1 ) and BPI(ω 1 ) fail in N .…”

Hausdorff compactifications in ZF

“…Clearly, if in a ZF-model M there do not exist free ultrafilters in the collection P(ω), then S(ω) = W is discrete in M, and the Alexandroff compactification α a W is a Hausdorff compactification of W but, in view of Theorem 2.27, α a W is not a Čech-Stone compactification of W because W is not amorphous. In the model N [Γ] of [11], S(ω) is a dense-in-itself Tychonoff space which is easily seen not to be locally compact. Hence, the Alexandroff compactification α a S(ω) of S(ω), in contrast to that of ω, is not Hausdorff in N [Γ].…”

“…That the Wallman space W(X, Z(X)) is compact for every Tychonoff space X is an equivalent of UFT (see, e.g., Theorem 2.8 of [22]). Some relatively new results on Hausdorff compactifications in ZF have been obtained in [14], [10], [11], [18], [19] and, for Delfs-Knebusch generalized topological spaces (applicable to topological spaces), in [22]. In [1], there is a well-written chapter on a history of Hausdorff compactifications in ZFC (see [3]).…”

On the metric reflection of a pseudometric space in ZF

“…Put X = A and let d : X × X → R be the pseudometric given by (3). We show that X is Weierstrass-compact.…”