Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9516987 | Topology and its Applications | 2005 | 13 Pages |
Abstract
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 (XâXâ1â{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 nâN, 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 inâ1(Uâ©Fn(X)) is open in (XâXâ1â{e})n for each n; (iii) X is locally compact separable or discrete.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Geometry and Topology
Authors
Kohzo Yamada,