(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.

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