Two weak forms of countability axioms in free topological groups

Given a Tychonoff space X  , let F(X)F(X) and A(X)A(X) be respectively the free topological group and the free Abelian topological group over X   in the sense of Markov. For every n∈Nn∈N, let Fn(X)Fn(X) (resp. An(X)An(X)) denote the subspace of F(X)F(X) (resp. A(X)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)F(X) or A(X)A(X), namely the csf-countability and snf-countability. We provide some characterizations of the csf-countability and snf  -countability of F(X)F(X) and A(X)A(X) for various classes of spaces X. In addition, we also study the csf-countability and snf  -countability of Fn(X)Fn(X) or An(X)An(X), for n=2,3,4n=2,3,4. Some results of Arhangel'skiı̌ in [1] and Yamada in [20] are generalized. An affirmative answer to an open question posed by Li et al. in [11] is provided.