Continuity in right semitopological groups

TL;DR

This paper investigates the properties and conditions under which right semitopological groups can be characterized as topological groups, expanding understanding of the relationship between algebraic and topological structures.

Contribution

It introduces new criteria and methods for analyzing when right semitopological groups are topological groups, using semi-neighborhoods of the diagonal.

Findings

Conditions under which right semitopological groups are topological groups

Use of semi-neighborhoods of the diagonal in analysis

Extension of classical results by Montgomery and Ellis

Abstract

Groups with a topology that is in consistent one way or another with the algebraic structure are considered. Classical groups with a topology are topological, paratopological, semitopological, and quasitopological groups. We also study other ways of matching topology and algebraic structure. The minimum requirement in this paper is that the group is a right semitopological group (such groups are often called right topological groups). We study when a group with a topology is a topological group; research in this direction began with the work of Deane Montgomery and Robert Ellis. (Invariant) semi-neighborhoods of the diagonal are used as a means of study.