Quasi-pseudometrics on quasi-uniform spaces and quasi-metrization of topological monoids

TL;DR

This paper introduces a new method for constructing quasi-pseudometrics on rotund quasi-uniform spaces, leading to new metrizability results for topological monoids and resolving an open problem from 2001.

Contribution

It presents a novel direct construction of right-continuous quasi-pseudometrics on rotund quasi-uniform spaces and applies it to topological monoids with open shifts, solving an open problem.

Findings

Topological monoids with open shifts are generated by right-continuous left-subinvariant quasi-pseudometrics.

A topological monoid with open shifts is completely regular iff it is semiregular.

The new construction simplifies proofs of classical metrizability theorems.

Abstract

We define a notion of a rotund quasi-uniform space and describe a new direct construction of a (right-continuous) quasi-pseudometric on a (rotund) quasi-uniform space. This new construction allows to give alternative proofs of several classical metrizability theorems for (quasi-)uniform spaces and also obtain some new metrizability results. Applying this construction to topological monoids with open shifts, we prove that the topology of any (semiregular) topological monoid with open shifts is generated by a family of (right-continuous) left-subinvariant quasi-pseudometrics, which resolves an open problem posed by Ravsky in 2001. This implies that a topological monoid with open shifts is completely regular if and only if it is semiregular. Since each paratopological group is a topological monoid with open shifts these results apply also to paratopological groups.