Complete Regularity: Kopperman's duality {\it \`{a} la quantale}

TL;DR

This paper refines Kopperman's duality by developing V-spaces, linking topological notions with metric counterparts, and reconstructing classical results through this new framework.

Contribution

It introduces a modern V-space theory that strengthens the duality between topological and metric structures, extending Kopperman's results.

Findings

Reconstruction of Kopperman's equivalence using V-spaces

Extension of classical topological results via metric framework

Development of a refined duality theory for topological spaces

Abstract

Nearly three decades from his celebrated result, we study a modern refinement and strengthening of Kopperman's full metrisabilty of all topological spaces. Within this new theory of \emph{V-spaces}, developed by Flagg and Weiss, we investigate several topological notions and their metric counterpart. Among our main results is the reconstruction, in terms of V-spaces, of Kopperman's equivalence between symmetric value semigroups and completely regular topologies. We conclude our work by revisiting some classical topological results and their almost evident validity through this metric lens.