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Topological spaces of persistence modules and their properties

  • Peter BubenikEmail author
  • Tane Vergili
Article
  • 88 Downloads

Abstract

Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules, including many of those that have been previously studied, and describe the relationships between them. In the cases where these classes are sets, interleaving distance induces a topology. We undertake a systematic study the resulting topological spaces and their basic topological properties.

Keywords

Persistent homology Persistence modules Interleaving distance 

Mathematics Subject Classification

55N99 54D99 54E99 18A25 

Notes

Acknowledgements

The authors would like to that the anonymous referees for their helpful suggestions. In particular, we would like to thank the referee who contributed the proof that the enveloping distance from pointwise-finite dimensional persistence modules to q-tame persistence modules is zero. We also thank Alex Elchesen for proofreading an earlier draft of the paper. The first author would like to acknowledge the support of UFII SEED funds, ARO Research Award W911NF1810307, and the Southeast Center for Mathematics and Biology, an NSF-Simons Research Center for Mathematics of Complex Biological Systems, under National Science Foundation Grant No. DMS-1764406 and Simons Foundation Grant No. 594594.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Supplementary material

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA
  2. 2.Department of MathematicsEge UniversityIzmirTurkey

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