Two weak forms of countability axioms in free topological groups

摘要

Given a Tychonoff space X, let F(X) and A(X) be respectively the free topological group and the free Abelian topological group over X in the sense of Markov. For every n is an element of N, let F-n,(X) (resp. A(n)(X)) denote the subspace of F(X) (resp. A(X)) that consists of words of reduced length at most n with respect to the free basis X. In this paper, we discuss two weak forms of countability axioms in F(X) or A(X), namely the csf-countability and snf-countability. We provide some characterizations of the csf-countability and snf-countability of F(X) and A(X) for various classes of spaces X. In addition, we also study the csf-countability and snf-countability of F-n(X) or A(n)(X), for n = 2,3,4. Some results of Arhangel'skii in [1] and Yamada in [20] are generalized. An affirmative answer to an open question posed by Li et al. in [11] is provided. (C) 2016 Elsevier B.V. All rights reserved.