Compact complement topologies and k-spaces

TL;DR

This paper investigates properties of compact complement topologies on Hausdorff spaces within ZF set theory, linking them to $k$-spaces, and explores their implications for various topological statements and axioms.

Contribution

It characterizes $k$-spaces using compact complement topologies and establishes equivalences involving the axiom of countable multiple choice within ZF.

Findings

Equivalence of CMC with all Hausdorff first countable spaces being $k$-spaces.

Construction of a countable metrizable space with a non-first countable compact complement topology.

Analysis of compact complement topologies in relation to Sorgenfrey line and Delfs-Knebusch generalized topologies.

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.