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.
