On the countable tightness and the k-property of free topological groups over generalized metrizable spaces

In this paper, the countable tightness and the k-space property in free topological groups over generalized metrizable spaces are considered. The following main results are obtained:(1) A space X   is separable or discrete if and only if the free topological group F(X)F(X) is of countable tightness for a paracompact σ-, k-space X;(2) A space X   is an ℵ0ℵ0-space or discrete if and only if F(X)F(X) is of countable tightness for a normal k⁎k⁎-metrizable k-space X;(3) A space X   is a kωkω-space or discrete if and only if F(X)F(X) is a k  -space for a k⁎k⁎-metrizable space X.Some results on the countable tightness and the k-property of free topological groups over metric spaces are improved.