In this paper, we prove that a space $X$ is a $g$-metrizable space if and only if $X$ is a weak-open, $\pi$ and $\sigma$-image of a semi-metric space, if and only if $X$ is a strong sequence-covering, quotient, $\pi$ and $mssc$-image of a semi-metric space, where “semi-metric” can not be replaced by “metric”.

In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We...