Notes on countable tightness of the subspaces of free (Abelian) topological groups

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 mainly discuss the subspaces F-n(X) and A(n)(X) with countable tightness for a Lasnev space X, and prove that: (1) Assume b = omega(1). For a non-metrizable Lasnev space X, the tightness of F-5 (X) is countable if and only if the tightness of F(X) is countable; (2) Let X be the closed image of a locally separable metrizable space. Then the tightness of A(4) (X) is countable if and only if the tightness of A(X) is countable. (C) 2017 Elsevier B.V. All rights reserved.