Cell complex

In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex).

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 n-dimensional open ball. A 0-dimensional open (and closed) cell is a singleton space. Closure-finite means that each closed cell is covered by a finite union of open cells.

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:

A CW complex is called regular if for each n-dimensional open cell C in the partition of X, the continuous map f from the n-dimensional closed ball to X is a homeomorphism onto the closure of the cell C.

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