Suppose we have a simplicial group $G$. What do we need from $G$ to get first countable $BG$, where $BG$ is a geometric realization of $G$?
A CWcomplex is first countable if and only if it's locally finite, and the geometric realization of a simplicial set is locally finite if and only if the original simplicial set was, in that only finitely many nondegenerate simplices may meet any other nondegenerate simplex. I don't see much improvement in this result from starting with a simplicial group.


1$\begingroup$ Not so much improvement but a shortcut  for a group, it should suffice to check the condition only for simplices containing one fixed 0simplex. $\endgroup$ – მამუკა ჯიბლაძე Dec 30 '18 at 6:20