Overview
CW complexes are way to build topological spaces from smaller pieces that layer on top of each other in increasing dimension. In particular, we use spheres and disks to do this.
Definition/ algorithm to build CW complex
- Start with a discrete set
. These are the 0-cells - Build the
-skeleton from inductively. We do this by “attaching” -cells to . This is done with maps
Note that
where we “glue” the boundary to the existing one by
where
- Stop (if you want) with a finite number
of skeleton and . Otherwise, continue infinintly with . In this case, has the weak topology where is open iff is open for all .
A topological space
Characteristic map
The cell complex is defined by the attaching maps
Each cell
that extends
where