Proximity spaces and topological functors

Abstract:

The purpose of this paper is to determine what natural functors $T:{\mathbf {A}} \to {\mathbf {X}}$ are $(\mathcal {E},\mathfrak {M})$-topological, where ${\mathbf {A}}$ is a subcategory of the category of proximity or uniform spaces and ${\mathbf {X}}$ is an $(\mathcal {E},\mathfrak {M})$-category. We give necessary and sufficient conditions under which a point separating family of continuous functions can be nicely lifted to a proximally continuous family. Proximities having a finest compatible uniform structure are characterized.