CW-complex

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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