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

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 h...

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Bibliographic Details
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
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Summary: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.
ISSN:0166-8641
1879-3207
DOI:10.1016/j.topol.2021.107732