CW 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. The idea was to have a class of spaces that was broader than simplicial complexes (we could say now, had better categorical properties); but still retained a combinatorial nature, so that computational considerations were not ignored. The name itself is unrevealing: CW stands for closure-finite weak topology.

CW complexes are defined inductively

Assume that X is to be a Hausdorff space: for the purposes of homotopy theory this loses nothing important. Then since closed cells are compact spaces, we can be sure that their images in X are also compact, closed subspaces. From now on, we refer to 'closed cells', and 'open cells', as subspaces of X, the open cell being the image of the distinguished interior.

Related Topics:
Hausdorff space - Compact space

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A 0-cell is just a point; if we only have 0-cells building up a Hausdorff space, it must be a discrete space. The general CW-complex definition can proceed by induction, using this as the base case.

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The first restriction is the closure-finite one: each closed cell should be covered by a finite union of open cells.

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The other restriction is to do with the possibility of having infinitely many cells, of unbounded dimension. The space X will be presented as a limit of subspaces Xi for i = 0, 1, 2, 3, … . How do we infer a topological structure for X? This is a colimit, in category theory terms. From the continuity of each mapping Xi to X, a closed set in X must have a closed inverse image in each Xi; and so must intersect each closed cell in a closed subset. We can turn this round, and say that a subset C of X is by definition closed precisely when the intersection of C with the closed cells in X is always closed.

Related Topics:
Colimit - Category theory

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With all those preliminaries, the definition of CW-complex runs like this: given X0 a discrete space, and inductively constructed subspaces Xi obtained from Xi−1 by attaching some collection of i-cells, the resulting colimit space X is called a CW-complex provided it is given the weak topology, and the closure-finite condition is satisfied for its closed cells.

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