二次可数紧T2空间作为$\mathbf{ZF}$中的补元和两个新的无限性概念
Second-countable compact Hausdorff spaces as remainders in $\mathbf{ZF}$ and two new notions of infiniteness
摘要：
在没有选择公理的情况下，在$\mathbf{ZF}$中给出了局部紧Hausdorff空间有所有非空的第二可数紧T2空间作为逼近器的充要条件.在其他独立性结果中，Hatzenhuhler和Mattson在$\mathbf{ZFC}$中得到的以所有非空可度量化紧空间为赋形器的局部紧T2空间的刻画被证明是独立于$\mathbf{ZF}$的. Urysohn的度量化定理被推广为如下定理：任何允许一个基可表示为有限集的可数并的$T_3$ -空间是可度量化的。应用到解决问题的存在性有关的一些特殊的度量化紧在$\mathbf{ZF}$。引入了强滤子基无限集和二阶滤子基无限集的新概念，这两个概念都源于对紧化的研究。集合论和拓扑定义的新概念，并与某些已知的无限集的概念，它们的关系进行了调查$\mathbf{ZF}$。引入了一个新的置换模型，其中存在一个弱Dedekind有限的强滤子基无限集。本文的所有$\mathbf{ZFA}$独立性结果均可转移至$\mathbf{ZF}$。
英文摘要：
In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in $\mathbf{ZF}$. Among other independence results, the characterization of locally compact Hausdorff spaces having all non-empty metrizable compact spaces as remainders, obtained by Hatzenhuhler and Mattson in $\mathbf{ZFC}$, is proved to be independent of $\mathbf{ZF}$. Urysohn's Metrization Theorem is generalized to the following theorem: every $T_3$-space which admits a base expressible as a countable union of finite sets is metrizable. Applications to solutions of problems concerning the existence of some special metrizable compactifications in $\mathbf{ZF}$ are shown. New concepts of a strongly filterbase infinite set and a dyadically filterbase infinite set are introduced, both stemming from the investigations on compactifications. Set-theoretic and topological definitions of the new concepts are given, and their relationship with certain known notions of infinite sets is investigated in $\mathbf{ZF}$. A new permutation model is introduced in which there exists a strongly filterbase infinite set which is weakly Dedekind-finite. All $\mathbf{ZFA}$-independence results of this article are transferable to $\mathbf{ZF}$.
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