Characterization of a metrizable space $X$ such that $F_4(X)$ is Fr\'echet-Urysohn

TL;DR

This paper characterizes when the subset of the free topological group on a metrizable space, consisting of words of length up to four, is Fréchet-Urysohn, revealing a specific condition on the space's non-isolated points.

Contribution

It provides the first equivalent condition on a metrizable space for the free topological group's subset of words of length four to be Fréchet-Urysohn, linking it to the compactness of non-isolated points.

Findings

If the non-isolated points of X form a compact set, then F_4(X) is Fréchet-Urysohn.

F_3(X) is Fréchet-Urysohn if and only if F_4(X) is Fréchet-Urysohn for a metrizable space.

The paper completes the characterization for n=4, previously known for n=3 and n≥5.

Abstract

Let $F(X)$ be the free topological group on a Tychonoff space $X$. For all natural numbers $n$ we denote by $F_n(X)$ the subset of $F(X)$ consisting of all words of reduced length $\leq n$. In \cite{Y3}, the author found equivalent conditions on a metrizable space $X$ for $F_3(X)$ to be Fr\'echet-Urysohn, and for $F_n(X)$ to be Fr\'echet-Urysohn for $n\geq5$. However, no equivalent condition on $X$ for $n=4$ was found. In this paper, we give the equivalent condition. In fact, we show that for a metrizable space $X$, if the set of all non-isolated points of $X$ is compact, then $F_4(X)$ is Fr\'echet-Urysohn. Consequently, for a metrizable space $X$ $F_3(X)$ is Fr\'echet-Urysohn if and only if $F_4(X)$ is Fr\'echet-Urysohn.