When reading about classifying spaces and universal (principal) bundles, multiple definitions seem to occur.
One has the following possibilities (as far as I can see):
The universal $G$-bundle $EG\rightarrow BG$ has a contractible total space $EG$ but only classifies numerable principal $G$-bundles (Constructed for example using the Milnor construction)
The universal bundle has a weakly contractible total space $EG$ but classifies all principal $G$-bundles.
First of all I was wondering if this is indeed the case as there are lots of papers and lecture notes (and I read at least a dozen by now) which do not clearly state the difference?
Next I was also wondering over which types of space $X$ the principal bundles $P\rightarrow X$ are classified as here the literature is also sometimes a bit unclear. Some papers only state the theorems for CW-complexes or paracompact Hausdorff spaces while others state the theorems for general topological spaces.