Countable tightness and $\mathfrak G$-bases on Free topological groups

TL;DR

This paper investigates the properties of countable tightness and $rak G$-bases in free topological groups and free Abelian groups over Tychonoff spaces, providing characterizations for various classes of spaces.

Contribution

It offers new characterizations of countable tightness and $rak G$-bases in free topological groups and their subgroups for different classes of spaces.

Findings

Characterizations of countable tightness for free topological groups.

Conditions for $rak G$-bases in free topological groups.

Analysis of these properties in subgroups $F_n(X)$.

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 consider two topological properties of $F(X)$ or $A(X)$, namely the countable tightness and $\mathfrak G$-base. We provide some characterizations of the countable tightness and $\mathfrak G$-base of $F(X)$ and $A(X)$ for various special classes of spaces $X$. Furthermore, we also study the countable tightness and $\mathfrak G$-base of some $F_{n}(X)$ of $F(X)$.