On feebly compact paratopological groups

TL;DR

This paper explores the properties of feebly compact paratopological groups, providing conditions for when such groups are topological, constructing examples of non-topological compact-like groups, and analyzing their product and quotient behaviors.

Contribution

It offers new criteria for feebly compact paratopological groups to be topological, constructs novel examples, and studies their algebraic and topological properties using cone topologies.

Findings

Feebly compact paratopological groups are topological iff quasiregularity holds.

Every 2-pseudocompact paratopological group is feebly compact.

The product of feebly compact paratopological groups is feebly compact.

Abstract

We obtain many results and solve some problems about feebly compact paratopological groups. We obtain necessary and sufficient conditions for such a group to be topological. One of them is the quasiregularity. We prove that each $2$-pseudocompact paratopological group is feebly compact and that each Hausdorff $\sigma$-compact feebly compact paratopological group is a compact topological group. Our particular attention concerns periodic and topologically periodic groups. We construct examples of various compact-like paratopological groups which are not topological groups, among them a $T_0$ sequentially compact group, a $T_1$ $2$-pseudocompact group, a functionally Hausdorff countably compact group (under the axiomatic assumption that there is an infinite torsion-free abelian countably compact topological group without non-trivial convergent sequences), and a functionally Hausdorff second countable group sequentially pracompact group. We investigate cone topologies of paratopological groups which provide a general tool to construct pathological examples, especially examples of compact-like paratopological groups with discontinuous inversion. We find a simple interplay between the algebraic properties of a basic cone subsemigroup $S$ of a group $G$ and compact-like properties of two basic semigroup topologies generated by $S$ on the group $G$. We prove that the product of a family of feebly compact paratopological groups is feebly compact, and that a paratopological group $G$ is feebly compact provided it has a feebly compact normal subgroup $H$ such that a quotient group $G/H$ is feebly compact.