On pseudobounded and premeager paratopological groups

TL;DR

This paper explores special classes of paratopological groups defined by boundedness and meagerness properties, establishing their relationships and addressing open questions in the field.

Contribution

It investigates the relations between pseudobounded, $ ext{ω}$-pseudobounded, and premeager paratopological groups, providing new insights and answers to existing open questions.

Findings

Established connections between the classes of groups.

Answered open questions posed by Lin, Lin, and Sánchez.

Clarified the structure and properties of these specialized groups.

Abstract

Let $G$ be a paratopological group. Following F. Lin and S. Lin, we say that the group $G$ is pseudobounded, if for any neighborhood $U$ of the identity of $G$, there exists a natural number $n$ such that $U^n=G$. The group $G$ is $\omega$-pseudobounded, if for any neighborhood $U$ of the identity of $G$, the group $G$ is a union of sets $U^n$, where $n$ is a natural number. The group $G$ is premeager, if $G\ne N^n$ for any nowhere dense subset $N$ of $G$ and any positive integer $n$. In this paper we investigate relations between the above classes of groups and answer some questions posed by F. Lin, S. Lin, and S\'anchez.