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

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.