论文标题
可计数的免费集号和PFA的空间
Spaces of countable free set number and PFA
论文作者
论文摘要
本文的主要结果是，在PFA下，对于每个{\ em常规} space $ x $带有$ f（x）=ω$，我们有$ | x | \ le w（x）^ω$;特别是，$ w（x）\ le \ mathfrak {c} $暗示$ | x | \ le \ mathfrak {c} $。这补充了许多先前的结果，这些结果均可获得一致的示例，即使是紧凑的hausdorff spaces $ x $，$ f（x）=ω$，因此$ w（x）= \ mathfrak {c} $和$ | x | = 2^\ Mathfrak {C} $。 我们还表明，在此结果中，规律性不能削弱到Hausdorff，因为我们可以在ZFC中找到hausdorff space $ x $ at $ f（x）=ω$，因此$ w（x）= \ mathfrak {c} $和$ | x | | = 2^\ Mathfrak {C} $。实际上，此空间$ x $具有{\ em强烈的抗urysohn}（sau）属性，即任何两个无限的封闭套件都在$ x $ Intersect中，它比$ f（x）=ω$强得多。此外，$ x $中的任何非空开放式设置还具有$ 2^\ mathfrak {c} $，因此通过在没有隔离点的ZFC a sau空间中提供\ cite {jshssz}的主要问题之一。
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.
