Abstract cell complex: Difference between revisions

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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-discrete Hausdorff space in a computer. (The neighborhood of each point in such a space must have infinitely many points).

The book by [[Vladimir Antonovich Kovalevsky|V. Kovalevsky]] <ref>V. Kovalevsky: "Geometry of Locally Finite Spaces". Editing house Dr. Bärbel Kovalevski, Berlin 2008. {{ISBN|978-3-9812252-0-4}}.</ref> contains the description of the theory of [[locally finite space]]s which are a generalization of abstract cell complexes. A locally finite space ''S'' is a set of points where a subset of ''S'' is defined for each point ''P'' of ''S''. This subset containing a limited number of points is called the '''smallest neighborhood''' of ''P''. A binary neighborhood relation is defined in the set of points of the locally finite space ''S'': The element (point) ''b'' is in the neighborhood relation with the element ''a'' if ''b'' belongs to the smallest neighborhood of the element ''a''. New axioms of a locally finite space have been formulated, and it was proven that the space ''S'' is in accordance with the axioms only if the neighborhood relation is anti-symmetric and transitive. The neighborhood relation is the reflexive hull of the inverse bounding relation. It was shown that classical axioms of the topology can be deduced as theorems from the new axioms. 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. It was demonstrated 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 leading from any cell of the complex to the cell ''c''. The bounding path is a sequence of cells in which each cell bounds the next 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. Using the abstract cell complexes, efficient algorithms for tracing, coding and polygonization of boundaries, as well as for the edge detection, are developed and described in the book <ref>Kovalevsky, V., Image Processing with Cellular Topology, Springer 2021, ISBN 978-981-16-5771-9.</ref>