The extensions of paratopological groups

In this paper, we attempt to extend some three space properties in topological groups to paratopological groups. The following results are established: (1) metrizability of compact (resp., sequentially compact, countably compact) subsets is a three space property in the class of k-gentle paratopological groups; (2) let N be a second-countable topological subgroup of an Abelian paratopological group G  ; if the quotient paratopological group G/NG/N has a countable network, then so does G; (3) let G be an Abelian paratopological group and N a topological subgroup of G; if both N   and G/NG/N are first-countable, then so is G.