The cometrizability of generalized metric spaces

A topological space $X$ is cometrizable if it admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets. We study the relation of cometrizable spaces to other generalized metric spaces and prove that all $\mathsf{as}$-cosmic spaces are cometrizable. Also, we present an example of a regular countable space of weight $\omega_1$, which is not cometrizable. Under $\omega_1=\mathfrak c$ this space contains no infinite compact subsets and hence is $\mathsf{cs}$-cosmic. Under $\omega_1<\mathfrak p$ this countable space is Fr\'echet-Urysohn and is not $\mathsf{cs}$-cosmic.