(This is Example 4.11 in Hatcher’s book).
Cellular Approximation for Pairs: Every map of CW pairs can be deformed through maps to a cellular map .
What “map of CW pairs” mean, is that is a map from to , and the image of under is contained in . CW pair means that is a cell complex, and is a subcomplex.
First, we use the ordinary Cellular Approximation Theorem to deform the restriction to be cellular. We then use the Homotopy Extension Property to extend this to a homotopy of on all of . Then, use Cellular Approximation Theorem again to deform the resulting map to be cellular staying stationary on .
We use this to prove a corollary: A CW pair is n-connected if all the cells in have dimension greater than . In particular the pair is n-connected, hence the inclusion induces isomorphisms on for and a surjection on .
First we note that being n-connected means that the space is non-empty, path-connected, and the first n homotopy groups are trivial, i.e. for .
Proof: First, we apply cellular approximation to maps with , thus the map is homotopic to a cellular map of pairs . Since all the cells in have dimension greater than , the n-skeleton of must be inside . Therefore is homotopic to a map whose image is in , and thus it is 0 in the relative homotopy group . This proves that the CW pair is n-connected. Note that 0-connected means path-connected.
Consider the long exact sequence of the pair :
Since it is an exact sequence, the image of any map equals the kernel of the next. Thus, (since ). Thus is surjective. Since , the later terms in the long exact sequence are also 0, thus, the inclusion induces isomorphisms on for , since the first n homotopy groups all vanish.