Two weak forms of countability axioms in free topological groups (original) (raw)

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 ∈ N, let Fn(X) (resp. An(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 Fn(X) or An(X), for n = 2, 3, 4. Some results of Arhangel'skiǐ in [1] and Yamada in [22] are generalized. An affirmative answer to an open question posed by Li et al. in [11] is provided.