Three-space properties in paratopological groups

Abstract

We show that neither first-countable nor second-countable are three-space properties in the class of paratopological groups: we present a countable regular paratopological Abelian group H which contains a closed discrete subgroup F such that H/F is topologically isomorphic to the rational numbers with the Sorgenfrey topology and H is not first-countable. Also, we prove that if H is an invariant topological subgroup of a paratopological group G such that H is second-countable and G/H has countable network, then G has countable network as well (this answers a question posed in [12]). Hence if H is an invariant topological subgroup of a first-countable paratopological group G such that H and G/H are second-countable, then so is G.