A regular topological space $X$ is called

$\bullet$ *cosmic* if $X$ is a continuous image of a separable metrizable space;

$\bullet$ *[cometrizable][1]* if $X$ admits a weaker metrizable topology such that each point $x\in X$ has a (not necessarily open) neighborhood base consisting of metrically closed sets.

In 1989 [Gruenhage][1] proved that under PFA a cometrizable space is cosmic if and only if it contains no uncountable discrete subspace and no uncountable subspace of the Sorgenfrey line.

Can the cometrizability be moved to the right-hand part of this characterziation?

>**Question.** Is each cosmic space cometrizable? 


  [1]: http://www.ams.org/journals/tran/1989-313-01/S0002-9947-1989-0992600-5/S0002-9947-1989-0992600-5.pdf