$\aleph_0$-spaces and images of separable metric spaces

A space $X$ is an $\aleph_0$-space if and only if $X$ is a sequencecovering and compact-covering image of a separable metric space. It follows that a space $X$ is a $k$-and-$\aleph_0$-space if and only if $X$ is a sequencecovering and compact-covering, quotient image of a separable metric space.