S2 and the Fréchet property of free topological groups

Let F(X)F(X) denote the free topological group over a Tychonoff space X  , Fn(X)Fn(X) denote the subspace of F(X)F(X) that consists of all words of reduced length ≤n with respect to the free basis X for every non-negative integer n   and En(X)=Fn(X)∖Fn−1(X)En(X)=Fn(X)∖Fn−1(X) for n≥1n≥1. In this paper, we study topological properties of free topological groups in terms of Arens' space S2S2. The following results are obtained.(1) If the free topological group F(X)F(X) over a Tychonoff space X   contains a non-trivial convergent sequence, then F(X)F(X) contains a closed copy of S2S2, equivalently, F(X)F(X) contains a closed copy of SωSω, which extends [6, Theorem 1.6].(2) Let X   be a topological space and A={n1,...,ni,...}A={n1,...,ni,...} be an infinite subset of NN. If C=⋃i∈NEni(X)C=⋃i∈NEni(X) is κ  -Fréchet–Urysohn and contains no copy of S2S2, then X is discrete, which improves [15, Proposition 3.5].(3) If X is a μ  -space and F5(X)F5(X) is Fréchet–Urysohn, then X is compact or discrete, which improves [15, Theorem 2.4].At last, a question posed by K. Yamada is partially answered in a shorter alternative way by means of a Tanaka's theorem concerning Arens' space S2S2.