CW-Complex

A CW-complex is a topological space constructed by successively attaching closed disks in increasing dimension, with the topology specified below. It generalizes the notion of a simplicial complex (Hatcher 2002, pp. 5-6).

Start with a discrete space , whose points are called 0-cells. For each integer , attach a collection of -dimensional closed disks to by continuous maps from their boundaries, where is the -sphere. The resulting n-skeleton is the quotient space of the topological disjoint union of and the closed disks, identifying each boundary point with . The images of the interiors of the closed disks are the -cells, each homeomorphic to an -dimensional open disk. There may be infinitely many -cells (Hatcher 2002, p. 5).

Set . A subset is declared closed iff is closed in for every nonnegative integer . This condition specifies the topology on the union, including when the cells have unbounded dimension (Hatcher 2002, pp. 5 and 519).

For example, a circle has a CW-complex structure with one 0-cell and one 1-cell. Attach both endpoints of the closed interval to the same point; its interior is the 1-cell. Thus an attaching map need not be injective (Hatcher 2002, p. 6).

By the CW-approximation theorem, every topological space admits a weak homotopy equivalence from a CW-complex . This map also induces isomorphisms on singular homology and singular cohomology groups (Hatcher 2002, pp. 352-353 and 356). Whitehead's theorem states that a weak homotopy equivalence between CW-complexes is a homotopy equivalence (Hatcher 2002, p. 352).