We prove that a space $X$ has a $\sigma$-locally finite Lindelöf sn-network if and only if $X$ is a compact-covering compact and mssc-image of a locally separable metric space, if and only if $X$ is a sequentially-quotient $\pi$ and mssc-image of a locally separable metric space, where ``compact-covering'' (or ``sequentially-quotient'') can not be replaced by ``sequence-covering''. As an application, we give a new characterization of spaces with locally countable weak bases.