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

摘要

Let F(X) be the free topological group on a Tychonoff space X. For all natural number n we denote by F_n(X) the subset of F(X) consisting of all words of reduced length ≤ n, and by i_n the natural mapping from (X ⊕ X~(-1) ⊕ {e})~n to F_n(X). We prove that for a metrizable space X if F_n (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 F_n (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 i_n is a quotient mapping for each n; (ⅱ) a subset U of F(X) is open if i_n~(-1) (U ∩ F_n (X)) is open in (X ⊕ X~(-1) ⊕ {e})~n for each n; (ⅲ) X is locally compact separable or discrete.