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Reconstruction of compacta by finite approximations and inverse persistence

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Abstract

The aim of this paper is to show how the homeomorphism and homotopy types of compact metric spaces can be reconstructed by the inverse limit of an inverse sequence of finite approximations of the space. This recovering allows us to propose an alternative way to construct persistence modules from a point cloud.

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Notes

  1. The definition still holds if we replace F by a commutative ring with unity, obtaining then R modules \(M_i\), but we need this stronger condition for the Structure Theorem.

  2. See [33] and [37] for details.

  3. For more about this topology see  [4, 5].

  4. This is a kind of Mittag–Leffer property (see [29]) for these elements of the inverse limit.

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Authors and Affiliations

  1. Departamento de Matemática Aplicada y Estadística, Facultad de Ciencias Económicas y Empresariales, Universidad San Pablo-CEU, CEU Universities, Calle Julián Romea 23, 28003, Madrid, Spain

    Diego Mondéjar Ruiz

  2. Departamento de Álgebra, Geometría y Topología, Facultad de Matemáticas, Universidad Complutense de Madrid and Instituto de Matemática Interdisciplinar, Plaza Ciencias 3, 28040, Madrid, Spain

    Manuel A. Morón

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  1. Diego Mondéjar Ruiz
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  2. Manuel A. Morón
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Correspondence to Diego Mondéjar Ruiz.

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This work has been partially supported by the research project PGC2018-098321-B-I00 (MICINN). The first author has been also supported by the FPI Grant BES-2010-033740 of the project MTM2009-07030 (MICINN).

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Mondéjar Ruiz, D., Morón, M.A. Reconstruction of compacta by finite approximations and inverse persistence. Rev Mat Complut 34, 559–584 (2021). https://doi.org/10.1007/s13163-020-00356-w

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