Topological spaces of persistence modules and their properties

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.

arXiv comment32 pages, to appear in Journal of Applied and Computational Topology

References in corpus (8)

• The Theory of the Interleaving Distance on Multidimensional Persistence Modules
• A persistence landscapes toolbox for topological statistics
• Metrics for generalized persistence modules
• Categorification of persistent homology
• Generalized Persistence Diagrams
• Algebraic Stability of Zigzag Persistence Modules
• Categorified Reeb Graphs
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Cited by in corpus (10)

• The persistence landscape and some of its properties
• Embeddings of Persistence Diagrams into Hilbert Spaces
• Approximating Continuous Functions on Persistence Diagrams Using Template Functions
• Exact weights, path metrics, and algebraic Wasserstein distances
• Basic Metric Geometry of the Bottleneck Distance
• On the Barcode Entropy of Reeb Flows
• Multi-Parameter Persistence Modules are Generically Indecomposable
• Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams
• Topological and metric properties of spaces of generalized persistence diagrams
• Barcode entropy for Reeb flows on contact manifolds with Liouville fillings