College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P. R. China
In this paper, spaces with $\sigma$-locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or $g$-first countable spaces with $\sigma$-locally countable $cs$-networks).
AMSclassification. 54E99, 54C10.
Keywords. Weak-bases, $cs$-networks, $k$-networks, $g$-first countable spaces, weakly open mappings, msss-mappings.