Let $G$ be a topological group.
The geometric bar construction $BG = B_{\bullet}(pt, G, pt)$ together with $EG = B_{\bullet}(pt,G,G)$ and the map $EG \to BG$ yields the universal principal $G$-bundle at least, when the identity $e \in G$ is a closed cofibration (a condition that is often called *well-pointed*).

It is claimed in the book by Rudyak "On Thom spectra, Orientability and Cobordism" in theorem 1.65 (iii) that this still holds true, if $G$ is not well-pointed. Is this an error in the book or did I miss something?

I only had the chance to look up one of the references that Rudyak gives. In "Classifying spaces and fibrations" by May, it is stated in theorem 8.2 that $EG \to BG$ is a principal $G$-bundle in case $e \in G$ is non-degenerate basepoint.

I guess my question is:

How bad is the fibration $EG \to BG$ in the case that $G$ is not well-pointed. Does $BG$ still have the right homotopy type / weak homotopy type?

strict group? $\endgroup$1more comment