Abstract cell complex: Difference between revisions

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==History==

The idea of abstract cell complexes (also named abstract cellular complexes) relates to [[Johann Benedict Listing|J. Listing]] (1862) <ref> Listing J.: "Der Census räumlicher Complexe". ''Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen'', v. 10, Göttingen, 1862, S. 97–182. </ref> und [[Ernst Steinitz|E. Steinitz]] (1908) <ref> Steinitz E.: "Beiträge zur Analysis". ''Sitzungsbericht Berliner Mathematischen Gesellschaft'', Band. 7, 1908, S. 29–49.</ref>. Also A.W Tucker (1933) <ref> Tucker A.W.: "An abstract approach to manifolds", Annals Mathematics, v. 34, 1933, 191-243. </ref>, K. Reidemeister (1938) <ref> Reidemeister K.: "Topologie der Polyeder und kombinatorische Topologie der Komplexe". Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1938 (second edition 1953) </ref>, P.S. Aleksandrov (1956) <ref> Aleksandrov P.S.: Combinatorial Topology, Graylock Press, Rochester, 1956, </ref> as well as R. Klette und A. Rosenfeld (2004) <ref> Klette R. und Rosenfeld. A.: "Digital Geometry", Elsevier, 2004. </ref> have described abstract cell complexes. E. Steinitz has defined an abstract cell complex as <math> C=(E,B,dim)</math> 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 <math>B(a, b)</math>, then <math>dim(a)<dim(b)</math>.

E. Steinitz has defined an abstract cell complex as <math> C=(E,B,dim)</math> 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 <math>B(a, b)</math>, then <math>dim(a)<dim(b)</math>.

V. Kovalevsky (1989) <ref>Kovalevsky, V.: "Finite Topology as Applied to Image Analysis",''Computer Vision, Graphics and Image Processing'', v. 45, No. 2, 1989, pp. 141–161.</ref> described abstract cell complexes for 3D and higher dimensions. He also suggested numerous applications to image analysis. In his book (2008) <ref>http://www.geometry.kovalevsky.de.</ref> 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 manifold]]s and many algorithms useful for image analysis.