Let 
and 
be CW-complexes, and let 
be a continuous map. Then the cellular approximation
theorem states that any such 
is homotopic to a cellular
map. In fact, if the map 
is already cellular on a CW-subcomplex 
of 
, then the homotopy can be taken to be stationary on 
.
A famous application of the theorem is the calculation of some homotopy groups of 
-spheres 
. Indeed, let 
and bestow on both 
and 
their usual CW-structure, with one 
-cell, and one 
-cell, respectively one 
-cell. If 
is a continuous, base-point preserving map, then
by cellular approximation, it is homotopic to a cellular map 
. This map 
must map the 
-skeleton of 
into the 
-skeleton of 
, but the 
-skeleton of 
is 
itself, while the 
-skeleton of 
is the zero-cell, i.e., a point. This is because of the
condition 
. Thus 
is a constant map, whence 
.
