The Milgram-Steenrod construction of classifying spaces for topological groups (Q1128351)

The authors revisit Steenrod's reworking of Milgram's construction of the classifying space \(BG\) for the topological group \(G\). Instead of using the compactly generated category, the authors work in the larger category of pointed weak Hausdorff \(k\)-spaces, \(wHk(\text{Top}_*)\). The Milgram-Steenrod construction gives a functor \[ {\mathcal B}: \text{Top}Gr_*\cap wHk(\text{Top}_*)\to wHk(\text{Top}_*) \] sending the topological group \(G\) to the classifying space \(BG\). (There is also a total space function \({\mathcal E}\).) \({\mathcal B}\) preserves closed inclusions. \({\mathcal B}\) preserves proclusions. \({\mathcal B}\) is exact. And \({\mathcal B}\) has other nice properties.