Each regular paratopological group is completely regular

TL;DR

This paper proves that regular paratopological groups are completely regular by characterizing semiregular spaces via normal quasi-uniformities, resolving longstanding open problems in the theory of paratopological groups.

Contribution

It introduces a new characterization of regular spaces using normal quasi-uniformities and applies this to establish that all regular paratopological groups are completely regular.

Findings

Regular paratopological groups are completely regular.

First countable Hausdorff paratopological groups are functionally Hausdorff.

A natural uniformity on paratopological groups helps resolve open problems.

Abstract

We prove that a semiregular topological space $X$ is completely regular if and only if its topology is generated by a normal quasi-uniformity. This characterization implies that each regular paratopological group is completely regular. This resolves an old problem in the theory of paratopological groups, which stood open for about 60 years. Also we define a natural uniformity on each paratopological group and using this uniformity prove that each (first countable) Hausdorff paratopological group is functionally Hausdorff (and submetrizable). This resolves another two known open problems in the theory of paratopological groups.