Oscillator topologies on a paratopological group and related number invariants

TL;DR

This paper introduces oscillator topologies on paratopological groups, defines related invariants, and explores conditions under which these groups admit weaker Hausdorff group topologies, along with various structural properties and examples.

Contribution

It develops the concept of oscillator topologies and invariants, providing new criteria for paratopological groups to admit weaker topologies and presenting novel examples.

Findings

A Hausdorff paratopological group is weaker Hausdorff if 3-oscillating.

All collapsing, nilpotent, and SIN paratopological groups are 2-oscillating.

Totally bounded groups are countably cellular and have uncountable cofinality as precaliber.

Abstract

We introduce and study oscillator topologies on paratopological groups and define certain related number invariants. As an application we prove that a Hausdorff paratopological group $G$ admits a weaker Hausdorff group topology provided $G$ is 3-oscillating. A paratopological group $G$ is 3-oscillating (resp. 2-oscillating) provided for any neighborhood $U$ of the unity $e$ of $G$ there is a neighborhood $V\subset G$ of $e$ such that $V^{-1}VV^{-1}\subset UU^{-1}U$ (resp. $V^{-1}V\subset UU^{-1}$). The class of 2-oscillating paratopological groups includes all collapsing, all nilpotent paratopological groups, all paratopological groups satisfying a positive law, all paratopological SIN-group and all saturated paratopological groups (the latter means that for any nonempty open set $U\subset G$ the set $U^{-1}$ has nonempty interior). We prove that each totally bounded paratopological group $G$ is countably cellular; moreover, every cardinal of uncountable cofinality is a precaliber of $G$. Also we give an example of a saturated paratopological group which is not isomorphic to its mirror paratopological group as well as an example of a 2-oscillating paratopological group whose mirror paratopological group is not 2-oscillating.