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=14376