A CW-complex is a homotopy-theoretic generalization of the notion of a simplicial complex. A CW-complex is any space X which can be built by starting off with a discrete collection of points called X^0, then attaching one-dimensional disks D^1 to X^0 along their boundaries S^0, writing X^1 for the object obtained by attaching the D^1s to X^0, then attaching two-dimensional disks D^2 to X^1 along their boundaries S^1, writing X^2 for the new space, and so on, giving spaces X^n for every n. A CW-complex is any space that has this sort of decomposition into subspaces X^n built up in such a hierarchical fashion (so the X^ns must exhaust all of X). In particular, X^n may be built from X^(n - 1) by attaching infinitely many n-disks, and the attaching maps S^(n - 1)->X^(n - 1) 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.

CW-approximation theorem | homology group | homotopy group | simplicial complex | space | subspace | Whitehead's theorem