可数自由集数空间与PFA
Spaces of countable free set number and PFA
摘要：
本文的主要结果是，在PFA下，对任何{ \em正则}空间$X$，有$F(X) = ω$，有$|X| \le w(X)^ω$，特别地，$w(X) \le \mathfrak{c}$蕴涵$|X| \le \mathfrak{c}$ .这补充了许多先前的结果，产生一致的例子，甚至紧凑的豪斯多夫空间$X$与$F(X) = ω$，使$w(X) = \mathfrak{c}$和$|X| = 2^\mathfrak{c}$。我们还证明了在这个结果中正则性不能减弱为Hausdorff，因为我们可以在ZFC中找到一个具有$F(X) = ω$的Hausdorff空间$X$，使得$w(X) = \mathfrak{c}$和$|X| = 2^\mathfrak{c}$ .实际上，这个空间$X$具有{ \em强反Urysohn}（SAU）性质，即$X$中任意两个无限闭集相交，这比$F(X) = ω$强得多。此外，$X$中的任何非空开集也具有大小$2^\mathfrak{c}$，从而通过在ZFC中提供没有孤立点的SAU空间来回答\cite{JShSSz}的主要问题之一。
英文摘要：
The main result of this paper is that, under PFA, for every {\em regular} space $X$ with $F(X) = ω$ we have $|X| \le w(X)^ω$; in particular, $w(X) \le \mathfrak{c}$ implies $|X| \le \mathfrak{c}$. This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. We also show that regularity cannot be weakened to Hausdorff in this result because we can find in ZFC a Hausdorff space $X$ with $F(X) = ω$ such that $w(X) = \mathfrak{c}$ and $|X| = 2^\mathfrak{c}$. In fact, this space $X$ has the {\em strongly anti-Urysohn} (SAU) property that any two infinite closed sets in $X$ intersect, which is much stronger than $F(X) = ω$. Moreover, any non-empty open set in $X$ also has size $2^\mathfrak{c}$, and thus answers one of the main problems of \cite{JShSSz} by providing in ZFC a SAU space with no isolated points.
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