On Hausdorff operators in ZF $\mathsf {ZF}$

Pub Date : 2023-07-24 DOI:10.1002/malq.202300004

Kyriakos Keremedis, Eleftherios Tachtsis

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Abstract

A Hausdorff space $(X, T)$ is called effectively Hausdorff if there exists a function F—called a Hausdorff operator—such that, for every $x, y \in X$ with $x \neq y$ , $F (x, y) = (U, V)$ , where U and V are disjoint open neighborhoods of x and y, respectively. Among other results, we establish the following in $ZF$ , i.e., in Zermelo–Fraenkel set theory without the Axiom of Choice ( $AC$ ):

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关于ZF$\mathsf｛ZF｝中的Hausdorff算子$

如果存在称为豪斯多夫算子的函数F，则豪斯多夫空间（X，T）$（X，\mathcal｛T｝）$被有效地称为豪斯道夫，使得对于每个X，y∈X$X，y\in X$中X≠y$X\ne y$，F（x，y）=（U，V）$F（x）=（U，V）$，其中U和V分别是x和y的不相交的开邻域。在其他结果中，我们在ZF$\mathsf{ZF}$中，即在没有选择公理的Zermelo–Fraenkel集合论（AC$\mathsf{AC}$）中建立了以下结果：

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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