Compactly generated rectifiable spaces or paratopological groups

Abstract

A rectifiable space (or a paratopological group) $G$ is compactly generated if $G=~\langle K\rangle$ for some compact subset $K$ of $G$. In this paper, we mainly discuss compactly generated rectifiable spaces or paratopological groups. The main results are that: (1) each $\sigma$-compact metrizable rectifiable space containing a dense compactly generated rectifiable subspace is compactly generated; (2) a metriable rectifiable space is compactly generated if and only if it is $\sigma$-compact and finitely generated modulo open sets; (3) any $\sigma$-compact paratopological group can be embedded as a closed paratopological subgroup in some compactly generated paratopological group. Finally, we consider generalized metric properties of compactly generated rectifiable spaces.