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 $$(X,E,\Phi)$$ where:


 * $$X$$ is a Hausdorff space
 * $$E$$ is a set of cells in $$X$$
 * $$\Phi$$ 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
 * $$(E,\Phi)$$ is called a CW-decomposition of $$X$$
 * $$\Phi_e$$ is termed the characteristic map of $$e$$