Compact complement topologies and k-spaces

Abstract

Let \((X,\tau)\) be a Hausdorff space, where \(X\) is an infinite set. The compact complement topology \(\tau^{\star}\) on \(X\) is defined by: \(\tau^{\star}=\{\emptyset\} \cup \{X\setminus M, \text{where \)M\( is compact in \)(X,\tau)\(}\}\). In this paper, properties of the space \((X, \tau^{\star})\) are studied in \(\mathbf{ZF}\) and applied to a characterization of \(k\)-spaces, to the Sorgenfrey line, to some statements independent of \(\mathbf{ZF}\), as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Among other results, it is proved that the axiom of countable multiple choice (\textbf{CMC}) is equivalent with each of the following two sentences: (i) every Hausdorff first countable space is a \(k\)-space, (ii) every metrizable space is a \(k\)-space. A \textbf{ZF}-example of a countable metrizable space whose compact complement topology is not first countable is given.