Corollary

Let G be a topological group, let B and $B'$ be paracompact topological spaces. Let $(E, p)$ and $(E', p')$ be principal fibre spaces with group G and with base B and $B'$ respectively. Suppose that the spaces E and $E'$ are both homotopic to a point. The spaces B and $B'$ are homotopic.

For there exists a continuous mapping $f: B\rightarrow B'$ such that the principal fibre space E is isomorphic to the principal fibre space $f^*E'$ and a continuous mapping $g: B'\rightarrow B$ such that the principal fibre space $E'$ is isomorphic to the principal fibre space $g^*E$ (Theorem 1, a)). The principal fibre spaces $(g\circ f)^*E$ and E are then isomorphic, therefore the mapping $g\circ f$ is homotopic to the mapping Id$_B$ (Theorem 1, b)). Analogously, the mapping $f\circ g$ is homotopic to the mapping Id$_{B'}$.

Let G be a topological group, let B be a topological space and let E be a principal fibre space with group G and with base B. Suppose that the space E is homotopic to a point. One says that the principal fibre space E is universal for principal fibre spaces with group G and paracompact base, and one says that the space B is a classifying space for G. If two classifying spaces for G are paracompact, they are homotopic. When there exists a classifying space, the study of isomorphism classes of principal fibre spaces with group G and paracompact base may be regarded as a homotopy problem.