Cellular approximation theorem

A continuous map between CW complexes is homotopic to a cellular map. If the map is already cellular on a subcomplex, the homotopy can be taken relative to that subcomplex. In particular, a map from a $d$-dimensional CW complex can be deformed into the $d$-skeleton of the target. This permits finite cellular chain complexes to compute induced homology maps, and turns maps to Complex projective space into maps to an appropriate finite-dimensional projective skeleton.