Classifying space
In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle E G → B G {\displaystyle EG\to BG} . As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces.