  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.
