Approach Theory in a Category: A Study of Compactness and Hausdorff Separation

Abstract

In a topological construct $\mathcal{X}$ endowed with a proper $(\mathcal{E}, \mathcal{M})$-factorization system and a concrete functor $\Lambda:\mathcal{X}\rightarrow \mathbf{Prap}$, we study $\mathcal{F}_{\Lambda}$-compactness and $\mathcal{F}_{\Lambda}$-Hausdorff separation, where $\mathcal{F}_{\Lambda}$ is a class of “closed morphisms” in the sense of Clementino et al. (A functional approach to general topology. In: Categorical Foundations. Encyclopedia of Mathematics and Its Applications, vol. 97, pp. 103–163. Cambridge University Press, Cambridge, 2004), determined by Λ. In particular, we point out under which conditions on Λ, the notion of $\mathcal{F}_{\Lambda}$-compactness of an object $\underline{X}$ of $\mathcal{X}$ coincides with 0-compactness of the image $\Lambda(\underline{X})$ in Prap. Our results will be illustrated by some examples: except for some well-known ones, like b-compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of b-compactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability.