aka cell complex

Motivation for CW Complexes – notes

See page 5 of Hatcher. CW Defn and Examples

See also Delta-complex

If $X=X^n$ for some $n$ then $X$ is finite dimensional and the smallest such $n$ is its dimension.

For a finite dimensional cell complex, we can define the Euler characteristic. the Euler characteristic of a cell complex depends only on its homotopy type, so the fact that the house with two rooms (from Hatcher) has the homotopy type of a point implies that its Euler characteristic must be 1, no matter how it is represented as a cell complex.

Each cell e α in a cell n complex X has a characteristic map $\Phi_\alpha:: D_\alpha^n \to X$ which extends the attaching map $\phi_\alpha$ and is a Homeomorphism from the interior of $D_\alpha^n$ onto $e_\alpha^n$ . Namely, we can take $\Phi_\alpha^n$ to be the composition $D^n_\alpha \hookrightarrow X^{n-1}\amalg_\alpha D_\alpha^n \to X^n \hookrightarrow X$ where the middle map is the quotient map defining $X^n$

So note that an n-cell is the interior of an n-Disk (see this video). The characteristic and attachment maps are defined on the disk corresponding to each n-cell.

A subcomplex is a closed subspace that is the union of cells in $X$. A pair (X, A) consisting of a cell complex X and a subcomplex A will be called a CW pair. For example, each skeleton X n of a cell complex X is a subcomplex.