Almost new pre-apartness from old

Abstract

Continuing the constructive theory of pre-apartness on frames, we investigate, in the context of an a-frame $\mathfrak{L}$ and its product $\mathfrak{P}$ with itself, a subset ${\mathfrak{B}}_w$ of $\mathfrak{P}$ whose counterpart in the context of a uniform space and the related apartness is closely linked with total boundedness. We also discuss the connection between the given pre-apartness on $\mathfrak{L}$ and a certain binary relation associated with ${\mathfrak{B}}_w$, also analogous to a binary relation on subsets of a uniform space.