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 ![[0,1]](/images/equations/CW-Complex/Inline21.svg)
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).
