snf-Countability and csf-countability in F-4(X)

摘要

Let F(X) be the free topological group on a Tychonoff space X, and F-n(X) the subspace of F(X) consisting of all words of reduced length at most n for each n is an element of N. In this paper conditions under which the subspace F-4(X) of the free topological group F(X) on a generalized metric space X contains no closed copy of S-omega 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 F-4 (X) is snf-countable if and only if X is compact or discrete; for a normal k- and N-space X F-4(X) is csf-countable if and only if X is an N-0-space or discrete; and for a k*-metrizable space X F-5(X) is a k -space and F-4(X) is csf -countable if and only if X is a k(omega)-space or discrete. Some results of K. Yamada, and F. Lin, C. Liu and J. Cao are improved. (C) 2017 Elsevier B.V. All rights reserved.