CW-Complex

A CW-complex is a homotopy-theoretic generalization of the notion of a simplicial complex. A CW-complex is any space  which can be built by starting off with a discrete collection of points called , then attaching one-dimensional disks  to  along their boundaries , writing  for the object obtained by attaching the s to , then attaching two-dimensional disks  to  along their boundaries , writing  for the new space, and so on, giving spaces  for every . A CW-complex is any space that has this sort of decomposition into subspaces  built up in such a hierarchical fashion (so the s must exhaust all of ). In particular,  may be built from  by attaching infinitely many -disks, and the attaching maps  may be any continuous maps.

The main importance of CW-complexes is that, for the sake of homotopy, homology, and cohomology groups, every space is a CW-complex. This is called the CW-approximation theorem. Another is Whitehead's theorem, which says that maps between CW-complexes that induce isomorphisms on all homotopy groups are actually homotopy equivalences.