First countable geometric realization of a simplicial group

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 $$G$$?

1 Answer

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.

• Thanks, Kevin. I will think about that – Fat ninja Nov 30 '18 at 7:59
• 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. – მამუკა ჯიბლაძე Dec 30 '18 at 6:20