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 $G$?
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A CW-complex 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 non-degenerate simplices may meet any other non-degenerate simplex. I don't see much improvement in this result from starting with a simplicial group.
answered Nov 30, 2018 at 1:22
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$\begingroup$ Thanks, Kevin. I will think about that $\endgroup$ Commented Nov 30, 2018 at 7:59
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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 0-simplex. $\endgroup$ Commented Dec 30, 2018 at 6:20