LAUSR

Abstract 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 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 = ω 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.