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).