Free Abelian topological groups and k-spaces

Kohzo Yamada, Shizuoka, Japan

Keywords: free topological group, free Abelian topological group, metrizable space, k-space, locally compact space AMS Subject classification: 22A05, 54D50, 54E45

Abstract: In this paper, we give a metrizable space X such that A_{n}(X) is a k-space for each n ∈ N, but A(X) is not a k-space. Moreover, we shall show that for a metrizable space X, if each F_{n}(X) (A_{n}(X)) is a k-space, then either X is locally compact or X′, the set of non-isolated points of X, is compact. On the other hand, we give a characterization of a metrizable space X such that every F_{n}(X) is locally compact.