# CW-complex

## Contents

## Inductive definition

A **CW-complex** is a topological space constructed inductively as follows. We start with the -skeleton, which is empty. The -skeleton, denoted , is constructed from the -skeleton , by attaching a discretely parametrized family of attaching maps from to , and taking the pushout with respect to these, for the inclusion of in .

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

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 where:

- is a Hausdorff space
- is a set of cells in
- is a family of maps parametrized by the members of

### Conditions

- is the disjoint union of all cells in
- For each -cell , the map is a relative homeomorphism
- The closure of any cell in is contained in a finite union of cells in
- has the weak topology determined by the closures of the cells in

### Terminology

- is termed a CW-space
- is called a
**CW-decomposition**of - is termed the characteristic map of