and on free topological groups

In this paper, we investigate copies of S ω and S 2 on free topological groups. By applying these results, we show that, for a paracompact space with a point-countable k-network, X is discrete or compact if F 5 ( X ) is Fréchet–Urysohn, which generalizes Yamadaʹs theorem (Yamada [26]). We also give a negative answer to Yamadaʹs conjecture (Yamada [26]): If X is a metrizable space, then F 4 ( X ) is Fréchet–Urysohn if and only if the set of all non-isolated points of X is compact; and a partial answer to Arhangelʹskiiʹs conjecture: S ω 1 cannot be embedded into a sequential topological group. Finally, we prove that, for a k ⁎ -metrizable μ-space X, the free topological group F ( X ) is a k-space if and only if F 5 ( X ) is a k-space. Some questions about free topological groups are posed.