Let (X, τ) be a Hausdorff space, where X is an infinite set. The compact complement topology τ on X is defined by: τ = {∅} ∪ {X M : M is compact in (X, τ)}. In this paper, properties of the space (X, τ) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (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 ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given.