The topological properties of $q$-spaces in free topological groups

TL;DR

This paper investigates the topological properties of free topological and Abelian groups over a space, focusing on when these groups exhibit $q$-space properties, thus addressing a specific open problem.

Contribution

It provides new insights into the conditions under which free topological groups and their finite levels are $q$-spaces, partially answering an existing open question.

Findings

Identifies conditions for $F(X)$ and $A(X)$ to be $q$-spaces

Establishes links between $q$-space properties and local $ extomega$-boundedness

Provides partial solutions to a problem in the topology of free groups

Abstract

Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. In this paper, we provide some topological properties of $X$ whenever one of $F(X)$, $A(X)$, some finite level of $F(X)$ and some finite level of $A(X)$ is $q$-space (in particular, locally $\omega$-bounded spaces and $r$-spaces), which give some partial answers to a problem posed in [11].