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## CW complexIn topology, aCW 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.
For these purposes a closed
## Attaching cells
A cell is attached by gluing a closed n-1-skeleton X_{n-1}, i.e., the union of all lower dimensional cells. The gluing is specified by a continuous function f from ∂D = ^{n}S^{n-1} to X_{n-1}. The points on the new space are exactly the equivalence classes of points in the disjoint union of the old space and the closed cell D, the equivalence relation being the transitive closure of ^{n}x≡f(x). The function f plays an essential role in determining the nature of the newly enlarged complex. For example, if D^{2} is glued onto S^{1} in the usual way, we get D itself; if ^{2}f has winding number 2, we get the real projective plane instead.## CW complexes are defined inductively
Assume that 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.
The first restriction is the
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
With all those preliminaries, the definition of CW-complex runs like this: given ## 'The' homotopy category
The idea of a homotopy category is to start with a topological space category, that is, one in which objects are topological spaces and morphisms are continuous mappings, and abstractly to replace the sets Map(
The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for
To a large extent the business of homotopy theory is to describe the homotopy category; in fact it turns out that calculating Hot( Auxiliary constructions may mean that spaces that are not CW complexes must be used on occasion, but half a century since Whitehead has left this definition of homotopy category in good shape. One basic result is that the representable functors on the homotopy category have a simple characterisation (Brown’s representability theorem). One important later development was that of spectra in homotopy theory, essentially the derived category idea in a form useful for topologists. | |||||||

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