CW Complex - Formulation

Formulation

Roughly speaking, a CW-complex is made of basic building blocks called cells. The precise definition prescribes how the cells may be topologically glued together. The C stands for "closure-finite", and the W for "weak topology".

An n-dimensional closed cell is the image of an n-dimensional closed ball under an attaching map. For example, a simplex is a closed cell, and more generally, a convex polytope is a closed cell. An n-dimensional open cell is a topological space that is homeomorphic to the open ball. A 0-dimensional open (and closed) cell is a singleton space.

A CW complex is a Hausdorff space X together with a partition of X into open cells (of perhaps varying dimension) that satisfies two additional properties:

• For each n-dimensional open cell C in the partition of X, there exists a continuous map f from the n-dimensional closed ball to X such that
  • the restriction of f to the interior of the closed ball is a homeomorphism onto the cell C, and
  • the image of the boundary of the closed ball is contained in the union of a finite number of elements of the partition, each having cell dimension less than n.
• A subset of X is closed if and only if it meets the closure of each cell in a closed set.