---
tag: 5SP8E66T9E
kind: definition
name: CW complex
msc: [57Q05, 55U10]
defines: 5W95QFHXPR
uses: [0MXB4CKHFW, 05DSV40EB9, 05GZ11PWRY, 0SY9ECQ19Q]
sources:
  - url: https://en.wikipedia.org/wiki/CW_complex
---

## Definition

Let $X$ be a [[Hausdorff space|0MXB4CKHFW]].
For $n \ge 0$ let $D^n$ be the closed unit [[ball|05DSV40EB9]] in $\R^n$, with interior $\mathring{D}^n$ and boundary $S^{n-1}$.
A *CW structure* on $X$ is a partition of $X$ into subsets $e_\alpha$, called *cells*, with extra data.
Each cell has a dimension $n_\alpha \ge 0$ and a [[continuous map|0SY9ECQ19Q]] $\Phi_\alpha \colon D^{n_\alpha} \to X$, its *characteristic map*.
These data satisfy:

1. $\Phi_\alpha$ restricts to a [[homeomorphism|05GZ11PWRY]] from $\mathring{D}^{n_\alpha}$ onto $e_\alpha$.
2. $\Phi_\alpha(S^{n_\alpha - 1})$ is contained in the union of finitely many cells $e_\beta$ with $n_\beta < n_\alpha$ (closure finiteness).
3. A subset $A \subseteq X$ is closed if and only if $\Phi_\alpha^{-1}(A)$ is closed in $D^{n_\alpha}$ for every $\alpha$ (weak topology).

A *CW complex* is a Hausdorff space with a CW structure.
The cell $e_\alpha$ is an $n_\alpha$-cell, and the union $X^n$ of all cells of dimension at most $n$ is the $n$-skeleton.

## Remarks

Equivalently, a CW complex is built inductively.
The skeleton $X^0$ is a discrete space, and $X^n$ is obtained from $X^{n-1}$ by attaching $n$-cells along maps $S^{n-1} \to X^{n-1}$.
The union $X = \bigcup_n X^n$ has the topology in which a set is closed when its intersection with each $X^n$ is closed.

Example: the sphere $S^n$ with $n \ge 1$ has a CW structure with one $0$-cell $\set{p}$ and one $n$-cell $S^n \setminus \set{p}$.
The characteristic map $D^n \to S^n$ collapses $S^{n-1}$ to $p$ and is a homeomorphism on the interior, so clause 1 holds.
Clause 2 holds since $\Phi(S^{n-1}) = \set{p}$, and clause 3 holds since $S^n$ is a quotient of $D^n \sqcup \set{p}$.

Non-example: the closed disk $D^2$ with one $0$-cell at each point of the boundary circle and one $2$-cell, the interior, fails clause 2.
The boundary of the $2$-cell meets infinitely many $0$-cells.

A CW complex is finite if it has finitely many cells; a finite CW complex is compact.