Free Abelian paratopological groups over metric spaces

Abstract

Let X be a metrizable space. Let FP(Y) and AP(X) be the free paratopological group over X and the free Abelian paratopological group over X, respectively. Firstly, we use asymmetric locally convex spaces to prove that if Y is a subspace of X then AP(Y) is topological subgroup of AP(X). Then, we mainly prove that:(a)if the tightness of AP(X) is countable then the set of all non-isolated points in X is separable;(b)if X is a z-space, then AP(X) is a k-space if and only if X is locally compact, locally countable and the set of all non-isolated points in X is countable;(c)AP2(X) is first-countable if and only if the set of all non-isolated points in X is finite.Moreover, we show that, for a Tychonoff space X, AP(X) has a countable k-network if and only if X is a countable space with a countable k-network. Finally, we give negative answers to three questions which were posed by Arhangel'skiı̌ and Tkachenko in [3]. Some questions concerned with free paratopological groups are posed.