The Milgram-Steenrod construction of classifying spaces for topological groups 
Renzo A. Piccinini & Mauro Spreafico 

A classifying space for a topological group G is the base space of a principal
G bundle (E_G, p_G, B_G) with (weakly) contractible total space E_G. 
Classifying spaces have been constructed using several different methods; 
amongst the most popular constructions we recall the Milnor construction, the
Dold-Lashof-Fuchs construction} and the Milgram-Steenrod contruction. The 
objective of this article is to review the contents of Steenrod's paper in the 
light of more streamlined techniques and, as a consequence, obtain new results 
about the classifying space functor B which transforms a topological group G 
into a space B_G. In fact, we shall prove that B is exact and preserves 
products, normality, closed inclusions, proclusions and closed cofibrations; 
moreover, we shall see that if Z is a central subgroup of a topological group 
G such that the pair (G,Z) is a Z -equivariant closed cofibration, the map 
B_q:B_G-->B_{G/Z} induced by the quotient map q:G-->G/Z gives rise to a 
locally trivial principal B_Z-bundle.