Cell complex
A separable space 
that is a union of non-intersecting cells. Here, by a 
-dimensional cell one means a topological space that is homeomorphic to the interior of the unit cube of dimension 
. If for each 
-dimensional cell 
of 
one is given a continuous mapping 
from the 
-dimensional cube 
into 
such that: 1) the restriction 
of 
to the interior 
of 
is one-to-one and the image 
is the closure 
in 
of 
(here 
is a homeomorphism of 
onto 
); and 2) the set 
, where 
is the boundary of 
, is contained in the union 
of the cells 
of 
, then 
is called a cell complex; the union 
is called the skeleton of dimension 
of the cell complex 
. An example of a cell complex is a simplicial polyhedron.
A subset 
of a cell complex 
is called a subcomplex if it is a union of cells of 
containing the closures of such cells. Thus, the 
-dimensional skeleton 
of 
is a subcomplex of 
. Any union and any intersection of subcomplexes of 
are subcomplexes of 
.
Any topological space can be regarded as a cell complex — as the union of its points, which are cells of dimension 0. This example shows that the notion of a cell complex is too broad; therefore narrower classes of cell complexes are important in applications, for example the class of cellular decompositions or CW-complexes (cf. CW-complex).
Cell complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cell_complex&oldid=46292