On P-bases and ωω-sn-networks

Abstract

In this paper, we discuss some properties of paratopological groups and coset spaces with P-bases and ωω-sn-networks, respectively. We mainly give the following results: (1) Let P=K(M) for a separable metric space M. If G is a Fréchet-Urysohn Hausdorff paratopological group having the property (⁎⁎) with a P-base, then G is first-countable, hence submetrizable; (2) Let P=K(M) for a separable metric space M. If H is a closed neutral subgroup of a topological group G such that G/H is a Fréchet-Urysohn space with a P-base, then G/H is first-countable, hence metrizable; (3) A Fréchet-Urysohn Hausdorff paratopological group G has an ωω-weak base if and only if it has an ωω-base; (4) Every Fréchet-Urysohn Hausdorff paratopological group having the property (⁎⁎) with an ωω-sn-network is first-countable, hence submetrizable.