$H$-closed quasitopological groups

TL;DR

This paper characterizes $H$-closed quasitopological groups, providing conditions for $H$-closedness, solving an open problem, and illustrating examples of non-compact $H$-closed groups.

Contribution

It offers a sufficient condition for $H$-closedness in quasitopological groups and characterizes $H$-closed topological groups as Ra
31kov-complete, solving a problem by Arhangel
31ski and Choban.

Findings

A sufficient condition for $H$-closedness in quasitopological groups.

Topological groups are $H$-closed iff they are Ra
31kov-complete.

Examples of non-compact $H$-closed quasitopological groups.

Abstract

An $H$-closed quasitopological group is a Hausdorff quasitopological group which is contained in each Hausdorff quasitopological group as a closed subspace. We obtained a sufficient condition for a quasitopological group to be $H$-closed, which allowed us to solve a problem by Arhangel'skii and Choban and to show that a topological group $G$ is $H$-closed in the class of quasitopological groups if and only if $G$ is Ra\v\i kov-complete. Also we present examples of non-compact quasitopological groups whose topological spaces are $H$-closed.