Abstract In this paper some properties of $csf$-countable spaces are discussed. $csf$-countable
spaces can be characterized by certain images of metrizable spaces, the first-countability of pesudocompact $csf$-countable spaces are explored, and it is proved a regular pseudo-compact paratopological
group is metrizable if and only if it is a $csf$-countable Frechet space
