In this paper we prove that a space $X$ is with a locally countable $sn$-network (resp., weak base) if and only if it is a compact-covering (resp., compact-covering quotient) compact and $ss$-image of a metric space, if and only if it is a sequentially-quotient (resp., quotient) $\pi$- and $ss$-image of a metric space, which gives a new characterization of spaces with locally countable $sn$-networks (or weak bases).