S2 and covering properties of free topological groups

In this paper, we consider topological properties of free topological groups in terms of Arens' space S2 and covering properties of free topological groups. We show that F4(X) contains no copy of S2 when the set of all non-isolated points of a metrizable X being compact. We also establish that Xn is submetrizable and subparacompact for each n∈ω if and only if F(X) is submetrizable and subparacompact which implies that the free topological group F(S) on Sorgenfrey line S is subparacompact. Moreover, suppose that X is a σ-discrete space if and only if F(X) σ-discrete which answers Open problem 7.6.16 in Arhangel'skiı̌ and Tkachenko (2008) [2].