Reconstruction of compacta by finite approximations and inverse persistence

Ruiz, Diego Mondéjar; Morón, Manuel A.

doi:10.1007/s13163-020-00356-w

Cited by 10 publications

(8 citation statements)

References 36 publications

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“…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%

“…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. In [18,19], it is proved a generalization of the result obtained in [6] to compact metric spaces. In [4], a similar result is obtained for topological spaces satisfying that are locally compact, paracompact and Hausdorff spaces, where Alexandroff spaces are considered.…”

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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“…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: Introductionmentioning

confidence: 99%

Computational approximations of compact metric spaces

Chocano,

Morón,

del Portal

2021

Preprint

Self Cite

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“…Recent research recognizes the critical role played by the theory of finite topological spaces in several fields of mathematics such as dynamical systems [6,16,9], group theory [5,4,11] algebraic topology (see [3,18] and the references given there) and geometric topology [21,10]. It is worth pointing out that important conjectures can be stated in terms of the theory of finite topological spaces.…”

Section: Introductionmentioning

confidence: 99%

“…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%

Shape of compacta as extension of weak homotopy of finite spaces

Chocano¹,

Morón²,

Portal³

2021

Preprint

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“…Recently, it has been proved that topological spaces can be approximated by Alexandroff spaces using the concepts of inverse systems (inverse sequences) and inverse limits, see [12] for compact metric spaces and [3] for locally compact, paracompact and Hausdorff spaces. One could ask if flows on a general space (X, T ) can be in some sense approximated by flows in sufficiently close elements in the corresponding approach of inverse systems mentioned above.…”

Section: Introductionmentioning

confidence: 99%