Fréchet-urysohn Spaces in Free Topological Groups

Let F (X) and A(X) be respectively the free topological group and the free Abelian topological group on a Tychonoff space X. For every natural number n we denote by Fn(X) (An(X)) the subset of F (X) (A(X)) consisting of all words of reduced length ≤ n. It is well known that if a space X is not discrete, then neither F (X) nor A(X) is Fréchet-Urysohn, and hence first countable. On the other hand, it is seen that both F2(X) and A2(X) are Fréchet-Urysohn for a paracompact Fréchet-Urysohn space X. In this paper, we prove first that for a metrizable space X, F3(X) (A3(X)) is Fréchet-Urysohn if and only if the set of all non-isolated points of X is compact and F5(X) is Fréchet-Urysohn if and only if X is compact or discrete. As applications, we characterize the metrizable space X such that An(X) is Fréchet-Urysohn for each n ≥ 3 and Fn(X) is Fréchet-Urysohn for each n ≥ 3 except for n = 4. In addition, however, there is a first countable, and hence Fréchet-Urysohn subspace Y of F (X) (A(X)) which is not contained in any Fn(X) (An(X)). We shall show that if such a space Y is first countable, then it has a special form in F (X) (A(X)). On the other hand, we give an example showing that if the space Y is Fréchet-Urysohn, then it need not have the