Abstract cell complex: Difference between revisions

Revision as of 09:02, 5 July 2025

Latest revision as of 09:04, 5 July 2025

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The topology of abstract cell complexes is based on a [[partial order]] in the set of its points or cells.

The notion of the abstract cell complex defined by E. Steinitz is related to the notion of an [[abstract simplicial complex]] and it differs from a [[simplicial complex]] by the property that its elements are not [[simplex|simplices]]: An ''n''-dimensional element of an abstract complexes mustdoes not necessarily have ''n''+1 zero-dimensional sides, and not each subset of the set of zero-dimensional sides of a cell is necessarily a cell. This is important since the notion of an abstract cell complexes can be applied to the two- and three-dimensional grids used in image processing, which is not true for simplicial complexes. A non-simplicial complex is a generalization which makes the introduction of cell coordinates possible: There are non-simplicial complexes which are Cartesian products of such "linear" one-dimensional complexes where each zero-dimensional cell, besides two of them, bounds exactly two one-dimensional cells. Only such Cartesian complexes make it possible to introduce such coordinates that each cell has a set of coordinates and any two different cells have different coordinate sets. The coordinate set can serve as a name of each cell of the complex which is important for processing complexes.

Abstract complexes allow the introduction of classical topology (Alexandrov-topology) in grids being the basis of digital image processing. This possibility defines the great advantage of abstract cell complexes: It becomes possible to exactly define the notions of connectivity and of the boundary of subsets. The definition of dimension of cells and of complexes is in the general case different from that of simplicial complexes (see below).