Compactly Generated Spaces and Quasi-spaces in Topology

Abstract

The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $$\textsf {Top}$$ of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class $$\mathcal {C}$$ of objects we generalize the notion of $$\mathcal {C}$$-generated spaces, from which we derive, for instance, a general concept of Alexandroff spaces. Furthermore, as done for $$\textsf {Top}$$, we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces.