-connectedness, finite approximations, shape theory and coarse graining in hyperspaces

Morón, Manuel A.; Cuchillo-Ibáñez, E.; Luzón, Ana

doi:10.1016/j.physd.2008.05.016

Cited by 10 publications

(9 citation statements)

References 38 publications

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“…We also give an alternative inverse sequence that reconstructs algebraic invariants. This inverse sequence is more suitable for computational reasons and answers positively the General Principle [1]. In addition, we study the relations of our inverse sequences with the Main Construction and the construction given in [6].…”

Section: It Is Not Difficult To Show the Following Two Propertiesmentioning

confidence: 88%

“…Given a compact metric space (X, d), we recall the Main Construction [1] or Finite Approximative Sequence (FAS) for X [19]. Definition 1.9.…”

Section: It Is Not Difficult To Show the Following Two Propertiesmentioning

confidence: 99%

“…2 Finite approximative sequences with the opposite order In this section, given a compact metric space (X, d), we construct an inverse sequence of finite T 0 topological spaces using the opposite order instead of the natural order used in [1] and [19]. From now on, if there is no explicit mention of the partial order considered on U 4 (A), where is a positive real value and A is an -approximation of X, then it is considered the partial order given as follows: C ≤ D if and only D ⊆ C. Let B(x, ) denote the open ball of center x ∈ X and radius .…”

Section: It Is Not Difficult To Show the Following Two Propertiesmentioning

confidence: 99%

“…Namely, for every compact polyhedron X there exists a natural inverse sequence of finite topological spaces such that its inverse limit contains a homeomorphic copy of X which is a strong deformation retract. The idea of using finite topological spaces for this purpose was earlier suggested for instance in [1], where it was introduced the so-called Main Construction and it was conjectured the General Principle. This principle states that the Main Construction can be used to extrapolate high dimensional topological properties of X, as in particular, the Čech homology groups in any dimension.…”

Section: Introductionmentioning

confidence: 99%

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Computational approximations of compact metric spaces

Chocano,

Morón,

del Portal

2021

Preprint

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Given a compact metric space X, we associate to it an inverse sequence of finite T 0 topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of X that is a strong deformation retract. We provide a method to approximate the homology groups of X and other algebraic invariants. Finally, we study computational aspects and the implementation of this method.Example 1.2. We consider the topological space X ⊆ R 2 given in Figure 1. Let U denote the open cover given by U 1 , U 2 , U 3 , U 4 and all the possible intersections of them. It is clear that N (U) has the same homotopy type of X, see Figure 1. This approach has a drawback. It is not always easy to find an open cover satisfying the hypothesis of the nerve theorem. Recently, interesting results have been obtained modifying

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Section: It Is Not Difficult To Show the Following Two Propertiesmentioning

confidence: 88%

“…Given a compact metric space (X, d), we recall the Main Construction [1] or Finite Approximative Sequence (FAS) for X [19]. Definition 1.9.…”

Section: It Is Not Difficult To Show the Following Two Propertiesmentioning

confidence: 99%

Section: It Is Not Difficult To Show the Following Two Propertiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computational approximations of compact metric spaces

Chocano,

Morón,

del Portal

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Recently, results about the reconstruction of compact metric spaces and its algebraic invariants point out that there are connections between shape theory and the theory of finite topological spaces (see [2,21,10]). Previously, some related results connecting shape theory, and some of its invariants, with finiteness can be found in [14,13].…”

Section: Introductionmentioning

confidence: 99%