Compact complement topologies and k-spaces

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{whereM$iscompactin$(X,\tau)}\}. In this paper, properties of the space $(X,{\tau }^{\star })$ are studied in $\mathbf{Z}\mathbf{F}$ and applied to a characterization of $k$-spaces, to the Sorgenfrey line, to some statements independent of $\mathbf{Z}\mathbf{F}$, 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 ($\mathbf{\text{CMC}}$) 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 $\mathbf{\text{ZF}}$-example of a countable metrizable space whose compact complement topology is not first countable is given.