Milnor classifying spaces for topological groups

Remark 9.2.17. More generally, for every topological group \(G\) there exists a topological space \(BG\) together with a principal \(G\)-bundle \(EG \to BG\) whose total space \(EG\) is contractible. In [Husemoller (1993), Section 4.12], it is shown that Milnor’s construction of \(BG\) classifies principal \(G\)-bundles on paracompact Hausdorff spaces. Since \(EG\) is contractible, Proposition 9.2.11 implies that the underlying anima of \(BG\) is the classifying anima \(\bB G\). Thus the terminology ‘classifying anima’ is compatible with the classical terminology ‘classifying space’.