Fréchet-Urysohn subspaces of free topological groups. III (Q1731337)

The author completes his classification, for metrizable spaces $X$, of when the subspace $F_n(X)$ (of words of length at most $n$) of the free topological group on $X$ is Fréchet-Urysohn. The space $F_2(X)$ is Fréchet-Urysohn [\textit{K. Yamada}, Topol. Proc. 23(Summer), 379--409 (1998; Zbl 0970.54032)], the space $F_3(X)$ is Fréchet-Urysohn iff the set of non-isolated points of $X$ is compact and for $n\ge 5$ the space $F_n(X)$ is Fréchet-Urysohn iff the space $X$ is compact or discrete [\textit{K. Yamada}, Proc. Am. Math. Soc. 130, No. 8, 2461--2469 (2002; Zbl 1006.54052)]. In the present paper it is shown that the space $F_4(X)$ is Fréchet-Urysohn if the set of non-isolated points of $X$ is compact. \par For the free abelian group the classification is simpler: $A_2(X)$ is always Fréchet-Urysohn and compactness of the set of non-isolated points is equivalent to the Fréchet-Urysohn property of $A_3(X)$, of $A_4(X)$, or of all $A_n(X)$. For Part I see [Topology Appl. 210, 81--89 (2016; Zbl 1350.54026)]. For Part II see [ibid. 283, Article ID 107342, 7 p. (2020; Zbl 1459.54023)].