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 no [[simplex | simplices]]: An ''n''-dimensional element of an abstract complexes must not have ''n''+1 zero-dimensional sides, and not each subset of the set of zero-dimensional sides of a cell is 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-simlicial 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 the complexes. Abstract complexes allow the introduction of classical topology (Aleksandrov-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).
The topology of abstract complexes is based on a [[partial order]] in the set of its points or cells. E. Steinitz has considered the asymmetric, irreflexive and transitive [[binary relation]] among the cells called '''bounding relation'''. He defined an abstract cell complex as: <math> C=(E, B, \dim) </math> where ''E'' is an abstract set of cells, ''B'' is the bounding relation and dim is a function assigning a non-negative integer number to each cell in such a way that if the cell ''a'' bounds the cell ''b'' then: <math> \dim(a)<\dim(b) </math>.
The notion of an abstract cell complex differs essentially from that of a CW-complex because an abstract cell complex is no [[Hausdorff space]]. This is important from the point of view of computer science since it is impossible to explicitly represent a non-discret Hausdorff space in a computer. (The neighborhood of each point in such a space must have infinitelly many points).
The notion of an abstract cell complex is closely related to that of an [[abstract simplicial complex]]. A non-simplicial complex is a generalization which allows the introduction of coordinates: there are non-simplicial complexs being Cartesian products of one-dimensional complexes. Only such complexes make it possible to introduce coordinates. An abstract cell complex whose cells look like squares in the 2D case or like cubes in the 3D case makes it possible to introduce coordinates of cells which is impossible in the case of simplicial complexs. Abstract complexes make also possible to introduce a classical topology (Aleksandrov topology) in grids which are the basis of digital image processing. This is the great advantage of abstract cell complexes: It becomes possible to exactly define the notions of connetedness and that of boundary of a subset in digital image processing. The dimension of cells and of complexes is in the general case defined differntly as for simplicial complexes (see below).
The book by V. Kovalevsky regards in his book <ref> V. Kovalevsky: "Geometry of Locally Finite Spaces". PublishingEditing Househouse Dr. BaerbelBärbel Kovalevski, Berlin 2008. ISBN 978-3-9812252-0-4. </ref> contains the theorydiscription of the theorie of '''locally finite spaces''' which are a generalization of abstract cell complexes. A locally finite space ''S'' is a set of points in whichwhere a subset of ''S'' is defined for each point ''P'' of ''S''. This subset containing a limited number of points andis called the '''smallest neighborhood''' is defined for each point of ''SP''. V. Kovalevsky defined theA binary '''neighborhood relation''' is defined in the set of points of athe locally finite space ''S'': The element (point) ''b'' is in the neighborhood relation towith the element ''a'' if ''b'' belongs to the smallest neighborhood of the element ''a''. HeNew formulated new axiomsaxiomes of a locally finite space have been formulated, and provedit was proven that the space ''S'' satisfiesis thein axiomsaccordance ifwith andthe axioms only if the neighborhoodneighbohood relation is antisymmetric and transitivetransitiv. The neighborhoodneighbohood relation is the reflexive hull of the inverseinvers bounding relation. HeIt showedwas shown that classical axiomsaxiomes of the topology can be deduced as theoremes from the new axioms as theoremsaxiomes. Therefore a locally finite space satisfying the new axioms is a particular case of a classical topological space. Its topology is a [[poset topology]] or [[Alexandrov topology]].
An abstract cell complex is a particular case of a locally finite space in which the dimension is defined for each point. V. KovalevskyIt haswas showndemonstrated that the dimension of a cell ''c'' of an abstract cell complex is equal to the length (number of cells minus 1) of the maximum bounding path (orleading neighborhoodfrom path)any leadingcell fromof ''c''the complex to athe cell ''c''<sup>0</sup>. whichThe bounding path is boundeda bysequence no otherof cells orin which doeseach notcell belong tobounds the smallestnext neighborhood of some cell different from ''c''one.
The book contains the theory of digital straight segments in 2D complexes, numerous algorithms for tracing boundaries in 2D and 3D, for economically encoding the boundaries and for exactly reconstructing a subset from the code of its boundary.
