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.

Properties

• The product of two CW complexes X and Y is itself a CW complex if at least one of them is locally finite i.e. it has a finite number of cells in each dimension.
• The function spaces Hom(X,Y) are not CW-complexes in general but are homotopic to CW-complexes by a theorem of John Milnor (1958). Actual function spaces occur in the somewhat larger category of compactly generated Hausdorff spaces.