The natural mappings in and k-subspaces of free topological groups on metrizable spaces

Let F(X) be the free topological group on a Tychonoff space X. For all natural number n we denote by Fn(X) the subset of F(X) consisting of all words of reduced length ⩽n, and by in the natural mapping from (X⊕X−1⊕{e})n to Fn(X). We prove that for a metrizable space X if Fn(X) is a k-space for each n, then X is locally compact and either separable or discrete. Therefore, as a corollary, we obtain that for a metrizable space X if Fn(X) is a k-space for all n∈N, then so is F(X). Furthermore, it is proved that for a metrizable space X the following are equivalent: (i) the mapping in is a quotient mapping for each n; (ii) a subset U of F(X) is open if in−1(U∩Fn(X)) is open in (X⊕X−1⊕{e})n for each n; (iii) X is locally compact separable or discrete.