Frechet-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 F-n(X) (A(n)(X)) the subset of F(X) (A(X)) consisting of all words of reduced length less than or equal to n. It is well known that if a space X is not discrete, then neither F(X) nor A(X) is Frechet-Urysohn, and hence first countable. On the other hand, it is seen that both F-2(X) and A(2)(X) are Frechet-Urysohn for a paracompact Frechet-Urysohn space X. In this paper, we prove first that for a metrizable space X, F-3(X) (A(3)(X)) is Frechet-Urysohn if and only if the set of all non-isolated points of X is compact and F-5(X) is Frechet-Urysohn if and only if X is compact or discrete. As applications, we characterize the metrizable space X such that An(X) is Frechet-Urysohn for each ngreater than or equal to3 and F-n(X) is Frechet-Urysohn for each ngreater than or equal to3 except for n=4. In addition, however, there is a first countable, and hence Frechet-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 Frechet-Urysohn, then it need not have the form.