Compactly generated rectifiable spaces or paratopological groups

A rectifiable space (or a paratopological group) G is compactly generated if G = < K > 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.