Abstract cell complex

In mathematics, an abstract cell complex (also abstract cell complex ) an abstract set of "cells" with a binary relation ( " included in the conclusion of" ) and a figure in the non-negative integers ( "Dimension "). The complex is called " abstract", because the "cells" no subsets of a Euclidean space are, as is the case with simplicial or CW - complexes. Abstract cell complexes play an important role in image analysis and in computer graphics.

Motivation

In the topology is commonly used (geometric ) cell complexes which are composed of ( open or closed ) cells, ie Subspaces homeomorphic to ( open or closed ) balls in Euclidean space. Usually, it is assumed that there is a CW complex. ( A more specific term are still simplicial complexes. ) Among other things, for applications in image processing, it is useful, instead of geometric cell complexes to be used abstractly defined cell complexes.

Definition

An abstract cell complex is given by

• A lot,
• A binary relation on
• A function,

Satisfy the following axioms:

• Follows and,
• The following.

As a rule, still following additional axiom is assumed by Tucker:

• If and, then there is a with and.

Several authors still use additional further axioms.

Elements of are referred to as cells. In the particular case of a geometrical cell complex is the dimension of the cell means that the cell is located in the end of the cell.

History

The idea of an abstract cell complex goes back to J. Listing (1862 ) and E. Steinitz (1908 ). Also AW Tucker ( 1933), Reidemeister (1938 ), Aleksandrov (1956 ) and R. Klette and A. Rosenfeld (2004 ) describe abstract cell complexes. Numerous work on image processing using abstract cell complexes, examples of this are the textbooks by Pavlidis, Rosenfeld and Serra. In Kovalevsky an axiomatic theory of locally finite topological spaces is proposed as a generalization of the abstract cell complexes.