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 Top of topological spaces and continuous functions, to study compactly generated spaces and quasi-spaces in this setting. Moreover, for a class C of objects we generalize the notion of C-generated spaces, from which we derive, for instance, a general concept of Alexandroff spaces. Furthermore, as done for Top, we also study, in our level of generality, the relationship between compactly generated spaces and quasi-spaces. y∈Y (r(x, y) ⊗ s(y, z)), and the order between V-relations is defined componentwise. There exists an involution given by transposition: for each r : X−→ Y , r • : Y −→ X is given by, for each (y, x) ∈ Y × X, r • (y, x) = r(x, y). Denoting the bottom element of the complete lattice V by ⊥, each map f : X → Y can be seen as a V-relation f : X−→ Y : for (x, y) ∈ X × Y , f (x, y) =    k, if f (x) = y ⊥, otherwise. Let T = (T, m, e) : Set → Set be a monad satisfying the Beck-Chevalley condition, (BC) for short, that is, T preserves weak pullbacks and the naturality squares of m are weak pullbacks [CHJ14].We fix a lax extension of T to V-Rel, again denoted by T, so that T : V-Rel → V-Rel is a lax functor