Spaces of countable free set number and PFA
Alan Dow, Istvan Juhasz

TL;DR
This paper investigates the size and weight of regular spaces with countable free set number under PFA, establishing bounds and providing counterexamples in ZFC, including a strongly anti-Urysohn space with no isolated points.
Contribution
It proves under PFA that regular spaces with countable free set number have size bounds related to their weight, and constructs a ZFC example of a Hausdorff SAU space with no isolated points.
Findings
Under PFA, regular spaces with F(X)=ω have size ≤ w(X)^ω.
Constructs a ZFC Hausdorff space with F(X)=ω, w(X)=c, and size 2^c.
Provides a ZFC example of a SAU space with no isolated points.
Abstract
The main result of this paper is that, under PFA, for every {\em regular} space with we have ; in particular, implies . This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces with such that and . We also show that regularity cannot be weakened to Hausdorff in this result because we can find in ZFC a Hausdorff space with such that and . In fact, this space has the {\em strongly anti-Urysohn} (SAU) property that any two infinite closed sets in intersect, which is much stronger than . Moreover, any non-empty open set in also has size , and thus answers one of the main problems of…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Economic theories and models · Mathematical and Theoretical Analysis
