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E-Connectedness, Finite Approximations, Shape Theory and Coarse Graining in Hyperspaces

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Morón, Manuel A. and Cuchillo Ibáñez, Eduardo and Luzón, Ana (2008) E-Connectedness, Finite Approximations, Shape Theory and Coarse Graining in Hyperspaces. Physica D-Nonlinear Phenomena, 237 (23). pp. 3109-3122. ISSN 0167-2789

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Official URL: http://www.sciencedirect.com/science/article/pii/S0167278908002261


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Abstract

We use upper semifinite hyperspaces of compacta to describe "-connectedness and to compute homology from finite approximations. We find another connection between "-
connectedness and the so called Shape Theory. We construct a geodesically complete R-tree, by means of "-components at different resolutions, whose behavior at infinite captures the topological structure of the space of components of a given compact metric space. We also construct inverse sequences of finite spaces using internal finite approximations of compact metric spaces.
These sequences can be converted into inverse sequences of polyhedra and simplicial maps by means of what we call the Alexandroff-McCord correspondence. This correspondence allows us to relate upper semifinite hyperspaces of finite approximation with the Vietoris-Rips complexes
of such approximations at different resolutions. Two motivating examples are included in the introduction. We propose this procedure as a different mathematical foundation for problems on data analysis. This process is intrinsically related to the methodology of shape theory. Finally this paper reinforces Robins’s idea of using methods from shape theory to compute homology from finite approximations.


Item Type:Article
Uncontrolled Keywords:E-connectedness; data analysis; Upper semifinite hyperspaces; Alexandroff-McCord correspondence; Vietoris-Rips complex; shape theory
Subjects:Sciences > Mathematics > Topology
ID Code:20312
Deposited On:07 Mar 2013 12:33
Last Modified:12 Dec 2018 15:13

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