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

Abstract

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.