Networks on free topological groups

In this paper, we investigate certain networks on free topological groups and free Abelian topological groups and obtain the following: (1) F(X) is a submetrizable, σ-space if and only if X is a submetrizable, σ-space; (2) Let X be a k⁎-metrizable, μ-space. F(X) is a k-space if and only if X is discrete or a k-space with a countable k-network consisting of compact subsets; A(X) is a k-space if and only if X is discrete or a k-space with a star-countable k-network consisting of compact subsets and the non-isolated point set is separable; (3) Suppose that X is a k⁎-metrizable, μ-space, then F4(X) is weakly first-countable if and only if X is discrete or compact; F(X) is a k-space if and only if F10(X) is a k-space.