Given a (Hausdorff) topological group , the Milnor construction of universal principal -bundles (also known as the Milnor’s join construction) constructs the infinite join of copies of , i.e. the colimit of joins
and canonically equipps with a continuous free right action of which permits a structure of a CW-complex such that the action of permutes its cells. Consequently, the natural projection is a model for the universal bundle of locally trivial principal -bundles over paracompact Hausdorff spaces, or equivalently, of numerable principal -bundles over all Hausdorff topological spaces.
wikipedia: classifying space
John W. Milnor, James Stasheff, Characteristic classes, Princeton Univ. Press
D. Husemöller, M. Joachim, B. Jurčo, M. Schottenloher, The Milnor construction: homotopy classification of principal bundles, doi, in: Basic Bundle Theory and K-Cohomology Invariants, Lecture Notes in Physics, 2008, vol. 726 (2008) 75-81