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In the absence of the Axiom of Choice, necessary and sufficient conditionsfor a locally compact Hausdorff space to have all non-empty second-countablecompact Hausdorff spaces as remainders are given in $\mathbf{ZF}$. Among otherindependence results, the characterization of locally compact Hausdorff spaceshaving all non-empty metrizable compact spaces as remainders, obtained byHatzenhuhler and Mattson in $\mathbf{ZFC}$, is proved to be independent of$\mathbf{ZF}$. Urysohn's Metrization Theorem is generalized to the followingtheorem: every $T_3$-space which admits a base expressible as a countable unionof finite sets is metrizable. Applications to solutions of problems concerningthe existence of some special metrizable compactifications in $\mathbf{ZF}$ areshown. New concepts of a strongly filterbase infinite set and a dyadicallyfilterbase infinite set are introduced, both stemming from the investigationson compactifications. Set-theoretic and topological definitions of the newconcepts are given, and their relationship with certain known notions ofinfinite sets is investigated in $\mathbf{ZF}$. A new permutation model isintroduced in which there exists a strongly filterbase infinite set which isweakly Dedekind-finite. All $\mathbf{ZFA}$-independence results of this articleare transferable to $\mathbf{ZF}$.

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