The regularity of quotient paratopological groups

TL;DR

This paper investigates the conditions under which quotient paratopological groups are Hausdorff and regular, establishing new criteria involving the group reflexion and providing a counterexample to regularity.

Contribution

It introduces new conditions for the regularity of quotient paratopological groups and constructs a counterexample demonstrating limitations of these conditions.

Findings

Quotients are Hausdorff and regular if H is closed and locally compact in G^flat.

Counterexample shows quotient can be Hausdorff but not regular.

Group reflexion plays a key role in quotient regularity.

Abstract

Let $H$ be a closed subgroup of a regular abelian paratopological group $G$. The group reflexion $G^\flat$ of $G$ is the group $G$ endowed with the strongest group topology, weaker that the original topology of $G$. We show that the quotient $G/H$ is Hausdorff (and regular) if $H$ is closed (and locally compact) in $G^\flat$. On the other hand, we construct an example of a regular abelian paratopological group $G$ containing a closed discrete subgroup $H$ such that the quotient $G/H$ is Hausdorff but not regular.