Universal principal bundles and classifying spaces

Let $G$ be a well-pointed topological group of CW type. All $G$-actions on principal bundles on this page are right actions. A classifying principal $G$-bundle is a numerable principal bundle

$p:EG⟶BG$

such that pullback induces a bijection

$[X,BG]\stackrel{\cong }{\to }\{\text{isomorphism classes of numerable principal }G\text{-bundles over }X\}$

for every CGWH space $X$. The space $BG$ is then a classifying space of $G$. A classifying bundle whose total space $EG$ is contractible is called a contractible universal model.

Contractibility of the total space is part of the model constructed below, but it is not by itself the definition of the displayed classification property for arbitrary bases. We will first construct Milnor's numerable principal bundle with contractible total space and then prove directly that it has the pullback property. The paracompact version requires a separate theorem saying that the locally trivial bundle under consideration is numerable; no such implication is built into this definition.

Choose $e_0\in EG$ over $b_0\in BG$. These points base the fiber sequence $G\to EG\to BG$, using $g\mapsto e_{0}g$ to identify its fiber with $G$.