Abstract cell complexes

In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point. The complex is called “abstract” since its points, which are called “cells”, are not subsets of a Hausdorff space as it is the case in Euclidean and CW complex. Abstract cell complexes play an important role in image analysis and computer graphics.

The idea of abstract cell complexes (also named abstract cellular complexes) relates to J. Listing (1862) und E. Steinitz (1908). Also A.W Tucker (1933), K. Reidemeister (1938), P.S. Aleksandrov (1956) as well as R. Klette und A. Rosenfeld (2004) have described abstract cell complexes. E. Steinitz has defined an abstract cell complex as ${\displaystyle C=(E,B,dim)}$ where E is an abstract set, B is an asymmetric, irreflexive and transitive binary relation called the bounding relation among the elements of E and dim is a function assigning a non-negative integer to each element of E in such a way that if ${\displaystyle B(a,b)}$, then ${\displaystyle dim(a)<dim(b)}$. V. Kovalevsky (1989) described abstract cell complexes for 3D and higher dimensions. He also suggested numerous applications to image analysis. In his book (2008) he has suggested an axiomatic theory of locally finite topological spaces which are generalization of abstract cell complexes. The book contains among others new definitions of topological balls and spheres independent of metric, a new definition of combinatorial manifolds and many algorithms useful for image analysis.

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