CW-complex

Inductive definition

A CW-complex is a topological space $X$ constructed inductively as follows. We start with the $-1$-skeleton, which is empty. The $n$-skeleton, denoted $X^n$, is constructed from the $(n-1)$-skeleton $X^{n-1}$, by attaching a discretely parametrized family of attaching maps from $S^{n-1}$ to $X^{n-1}$, and taking the pushout with respect to these, for the inclusion of $S^{n-1}$ in $D^n$.

The space $X$ is given the union topology for the ascending unions of the skeleta. Each $n$-skeleton is closed in $X$, but not necessarily open in $X$.

A somewhat more general notion than a CW-complex is a cell complex, where we again attach cells, but it is now possible to attach the cells in any order rather than in the order of increasing dimension.

Definition (assuming Hausdorffness)

A CW-complex is the following data, subject to the following conditions.

Data

An ordered triple $(X,E,\mathrm{\Phi })$ where:

• $X$ is a Hausdorff space
• $E$ is a set of cells in $X$
• $\mathrm{\Phi }$ is a family of maps parametrized by the members of $E$

Conditions

• $X$ is the disjoint union of all cells in $E$
• For each $k$-cell $e\in E$, the map $(D^k,S^{k-1})\to (e\cup X^{k-1},X^{k-1})$ is a relative homeomorphism
• The closure of any cell in $E$ is contained in a finite union of cells in $E$
• $X$ has the weak topology determined by the closures of the cells in $E$

Terminology

• $X$ is termed a CW-space
• $(E,\mathrm{\Phi })$ is called a CW-decomposition of $X$
• ${\mathrm{\Phi }}_e$ is termed the characteristic map of $e$