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==History==
The idea of abstract cell complexes (also named abstract cellular complexes) relates to J. Listing (1862) <ref>Listing J.: "Der Census raeumlicher Complexe". ''Abhandlungen der Koeniglichen Gesellschaft der Wissenschaften zu Goettingen'', v. 10, Goettingen, 1862, pp. 97–182.</ref> and E. Steinitz (1908).<ref>Steinitz E.: "Beitraege zur Analysis". ''Sitzungsbericht Berliner Mathematischen Gesellschaft'', v. 7, 1908, pp. 29–49.</ref> Also A.W Tucker (1933) <ref> Tucker A.W.: "An abstract approach to manifolds", Annals Mathematics, v. 34, pp. 191-243. </ref>, K. Reidemacher <ref> Reidemacher K.: "Topologie der Polyeder und kombinatorische Topologie der Komplexe". Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1038 (second edition 1953). </ref> as well as R. Klette and A. Rosenfeld (2004) <ref> Klette R. and Rosenfeld. A.: "Digital Geometry", Elsevier, 2004. </ref> have abstract cell complexes described. 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.
==Basic results==
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