# Cell complex

A separable space $ X $
that is a union of non-intersecting cells. Here, by a $ p $-
dimensional cell one means a topological space that is homeomorphic to the interior of the unit cube of dimension $ p $.
If for each $ p $-
dimensional cell $ t ^ {p} $
of $ X $
one is given a continuous mapping $ f $
from the $ p $-
dimensional cube $ I ^ {p} $
into $ X $
such that: 1) the restriction $ f ^ { * } $
of $ f $
to the interior $ \mathop{\rm Int} I ^ {p} $
of $ I ^ {p} $
is one-to-one and the image $ f ( I ^ {p} ) $
is the closure $ \overline{ {t ^ {p} }}\; $
in $ X $
of $ t ^ {p} $(
here $ f ^ { * } $
is a homeomorphism of $ \mathop{\rm Int} I ^ {p} $
onto $ t ^ {p} $);
and 2) the set $ f ( \partial I ^ {p} ) $,
where $ \partial I ^ {p} $
is the boundary of $ I ^ {p} $,
is contained in the union $ X ^ {p-} 1 $
of the cells $ t ^ {p-} 1 $
of $ X $,
then $ X $
is called a cell complex; the union $ X ^ {p-} 1 $
is called the skeleton of dimension $ p - 1 $
of the cell complex $ X $.
An example of a cell complex is a simplicial polyhedron.

A subset $ L $ of a cell complex $ X $ is called a subcomplex if it is a union of cells of $ X $ containing the closures of such cells. Thus, the $ n $- dimensional skeleton $ X ^ {n} $ of $ X $ is a subcomplex of $ X $. Any union and any intersection of subcomplexes of $ X $ are subcomplexes of $ X $.

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).

**How to Cite This Entry:**

Cell complex.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cell_complex&oldid=46292