Second-countable compact Hausdorff spaces as remainders in $\mathbf{ZF}$ and two new notions of infiniteness

TL;DR

This paper explores the conditions under which certain compact Hausdorff spaces appear as remainders in locally compact spaces within $ ext{ZF}$ set theory, introduces new notions of infiniteness, and generalizes Urysohn's Metrization Theorem.

Contribution

It provides necessary and sufficient conditions for remainders in $ ext{ZF}$, proves independence results related to these conditions, and introduces new concepts of infiniteness with applications to compactifications.

Findings

Characterization of remainders in $ ext{ZF}$

Independence of certain space characterizations from $ ext{ZF}$

Introduction of new infiniteness notions and their set-theoretic properties

Abstract

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 $\mathbf{ZF}$.