\(\varepsilon \)-connectedness, finite approximations, shape theory and coarse graining in hyperspaces
type: Item
instance of: scholarly article
title: \(\varepsilon \)-connectedness, finite approximations, shape theory and coarse graining in hyperspaces
zbMATH Open document ID: 1165.54005
author: Manuel A. Morón
author: Eduardo Cuchillo-Ibáñez
author: Ana Luzón
published in: Physica D
publication date: 2009-02-10T00:00:00Z
review text: As a starting point, this paper studies \(\varepsilon\)-connectedness. This concept was introduced by \textit{G. Cantor} [Math. Ann. 23, 453--488 (1884; JFM 16.0459.01)], and recently studied and applied by \textit{D. Snyder} [Physica D 173, No.~3--4, 131--136 (2002; Zbl 1007.54015)] and \textit{V. Robins}, \textit{J. D. Meiss} and \textit{E. Bradley} [Nonlinearity 11, No. 4, 913--922 (1998; Zbl 0957.54010)]. After establishing an important connection between \(\varepsilon\)-connectedness and shape theory, many related applications to computational topology and dynamics are stated. In this regard, the work of V. Robins and collaborators is noteworthy; references are provided.
Mathematics Subject Classification ID: 54C56
zbMATH DE Number: 5506281
zbMATH Keywords: \(\varepsilon \)-connectedness
zbMATH Keywords: data analysis
zbMATH Keywords: upper semifinite hyperspaces
zbMATH Keywords: Alexandroff-McCord correspondence
zbMATH Keywords: Vietoris-Rips complex
zbMATH Keywords: shape theory
reviewed by: Sukhjit Singh
MaRDI profile type: Publication
full work available at URL: j.physd.2008.05.016
OpenAlex ID: W2084230705
cites work: Computing connectedness: An exercise in computational topology
cites work: Computing connectedness: disconnectedness and discreteness.
cites work: Q4424951
cites work: Barcodes: The persistent topology of data
cites work: Finite approximations to Čech homology
cites work: Fundamental properties of \(\varepsilon\)-connected sets
cites work: Homotopical properties of upper semifinite hyperspaces of compacta
cites work: Upper semifinite hyperspaces as unifying tools in normal Hausdorff topology
cites work: Shapes for topological spaces
cites work: Q3959112
cites work: Shape theory. An introduction
cites work: Q5767704
cites work: Singular homology groups and homotopy groups of finite topological spaces
cites work: Finite Topological Spaces
cites work: Q4266107
cites work: THE HAUSDORFF METRIC AND CLASSIFICATIONS OF COMPACTA
cites work: Topologies on Spaces of Subsets
cites work: Q4277838
cites work: Q3869981
cites work: Concerning homotopy properties of compacta
cites work: Q4078819
cites work: Q4164073
cites work: Q3874535
cites work: Q3482537
cites work: Strong cellularity and global asymptotic stability
cites work: Shift equivalence in homotopy
cites work: Every Attractor of a Flow on a Manifold has the Shape of a Finite Polyhedron
cites work: Multihomotopy, Čech Spaces of loops and Shape Groups
cites work: On the structure of uniform attractors
cites work: On the global structure of invariant regions of flows with asymptotically stable attractors
cites work: Shape and Morse theory of attractors
cites work: Some duality properties of non-saddle sets
cites work: Morse equations and unstable manifolds of isolated invariant sets
cites work: Shape of global attractors in topological spaces
cites work: A note about the shape of attractors of discrete semidynamical systems
cites work: Topology and dynamics of unstable attractors
cites work: Pointed shape and global attractors for metrizable spaces
cites work: Global attractors: Topology and finite-dimensional dynamics
cites work: The shape of a cross-section of the solution funnel of an ordinary differential equation
cites work: Dynamical systems, shape theory and the Conley index
cites work: Q4312358
cites work: Q5520521
cites work: An Intrinsic Description of Shape
cites work: Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold
DOI: 10.1016/J.PHYSD.2008.05.016
description: scientific article; zbMATH DE number 5506281