Second-countable compact Hausdorff spaces as remainders in ZF and two new notions of infiniteness

• Published in: Topology and its applications 2021-07, Vol.298, p.107732, Article 107732
• Main Authors: Keremedis, Kyriakos, Tachtsis, Eleftherios, Wajch, Eliza
• Format: Article
• Language: English
• Subjects: Cantor set; Compactification; Metrizability; Remainder; Urysohn's metrization theorem; Weak forms of the Axiom of Choice
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2021.107732

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 Hatzenbuhler and Mattson in ZFC, is proved to be independent of ZF. Urysohn's Metrization Theorem is generalized. 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.