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 ≤n. In [9], the author found equivalent conditions on a metrizable space X for $F_3(X)$ to be Fréchet-Urysohn, and for $F_n(X)$ to be Fréchet-Urysohn for $n\ge 5$. However, no equivalent condition on X for $n=4$ was found. Recently, we proved in [11] that if a metrizable space X is locally compact separable and the set of all non-isolated points of X is compact, then $F_4(X)$ is Fréchet-Urysohn. In this paper we continue these investigations and show that ‘separability’ of X can be omitted. The result is an affirmative answer to Question 2 in [11] as well as Question 4.2 in [4].

让 $F（X）$是Tychonoff空间X上的自由拓扑群。对于所有自然数n，我们用$F_{ñ}（X）$ 的子集 $F（X）$由减小的长度≤的所有单词Ñ。在[9]中，作者发现等价条件上的度量化空间X为$F_3（X）$ 成为Fréchet-Urysohn，并且 $F_{ñ}（X）$ 成为Fréchet-Urysohn $ñ\ge 5$。然而，在没有相应的条件X为$ñ=4$被找到。最近，我们在[11]中证明，如果一个可量化的空间X是局部紧致可分离的，并且X的所有非孤立点的集合都是紧致的，则$F_4（X）$是Fréchet-Urysohn。在本文中，我们继续进行这些研究，并表明可以省略X的“可分离性” 。结果是对[11]中问题2和[4]中问题4.2的肯定回答。