2020/09/20 by Kyriakos Keremedis, Keremedis, Kyriakos, Eleftherios Tachtsis +3
Computer Science · Mathematics · #03E25 #03E35 #54A35 #54D35 #54D40 #54D45 #54E35 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2009.09526
openalex publication_date 2020/09/20 · openalex created_date 2020/09/25 · openalex updated_date 2026/07/28
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 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 ZFC, is proved to be independent of ZF. Urysohn's Metrization Theorem is generalized to the following theorem: every T3-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 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 ZF. A new permutation model is introduced in which there exists a strongly filterbase infinite set which is weakly Dedekind-finite. All ZFA-independence results of this article are transferable to ZF.