snf-Countability and csf-countability in F4(X)

Let F(X) be the free topological group on a Tychonoff space X, and Fn(X) the subspace of F(X) consisting of all words of reduced length at most n for each n∈N. In this paper conditions under which the subspace F4(X) of the free topological group F(X) on a generalized metric space X contains no closed copy of Sω are obtained and used to discuss countability axioms in free topological groups. It is proved that for a k-semistratifiable k-space X the subspace F4(X) is snf-countable if and only if X is compact or discrete; for a normal k- and ℵ-space X F4(X) is csf-countable if and only if X is an ℵ0-space or discrete; and for a k⁎-metrizable space X F5(X) is a k-space and F4(X) is csf-countable if and only if X is a kω-space or discrete. Some results of K. Yamada, and F. Lin, C. Liu and J. Cao are improved.