TITLE: Classifying spaces of topological groups that are not well-pointed
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: I don't have a real answer but possibly relevant observations and speculations. I imagine $BG$ can be pretty horrible if $G$ is not well-pointed, unless of course one uses the fat realization, when I imagine the fat construction of $EG$ does not give a bundle.  One remark is that $G$ is homotopy equivalent to a well-pointed monoid (not group) $G'$, as noted in Remarks 9.3 of Classifying spaces and fibrations.  One might try to compare the categories of principal $G$-fibrations and principal $G'$-fibrations.  In another direction, for a space to be well-pointed means that it is cofibrant in the Hurewicz (or Strom) model structure on based topological spaces.  As will probably be apparent, I haven't really thought about this, but one might ask for a Hurewicz model structure on topological groups for which cofibrant approximation gives a homotopy equivalent well-pointed group.