Brown's representability theorem and branched coverings

It is well known that Brown's Representability Theorem has many applications. The proof of the existence of a universal bundle for a topological group (see [4]) and the proof that any space can be approximated by CWcomplexes (see [8]) are among those applications. Here we prove the existence of classifying spaces for branched coverings over CW-complexes using Brown's Theorem as the main tool.