The cellular approximation theorem states that every continuous map between CW complexes (with chosen CW presentations) is homotopic to a cellular map? (a map induced by a map of cell complexes).

This is a CW analogue of the simplicial approximation theorem (sometimes also called lemma): that every continuous map between the geometric realizations of simplicial complexes is homotopic to a map induced by a map of simplicial complexes (after subdivision).

Statement

Theorem

Given a continuous map $f: (X, A) \to (X', A')$ between relative CW complexes that is cellular on a subcomplex $(Y, B)$ of $(X, A)$, there is a cellular map $g: (X, A) \to (X', A')$ that is homotopic to $f$ relative to $Y$.