Homological illusions of persistence and stability

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Wasserstein Stability for Persistence Diagrams (2020, unknown): “This corollary states that for X a triangulable, compact metric space that implies bounded degree-k total persistence for some real number k ≥ 1 , and f, g two tame Lipschitz functions we have See Section 3 for some limitations in terms of k and C X,k .”
Proximity of persistence modules and their diagrams (2009, background): “in [20], it can be replaced by some PL function with the same persistence diagram, defined over the first barycentric subdivision of C.”
Statistical Analysis and Parameter Selection for Mapper (2017, background): “ fon the geometric realization jKjof K. For a xed simplicial complex K, the extended persistence diagram of the lower-star ltration induced by f and of the sublevel sets of fPL are identical|see e.g. Morozov (2008), therefore the map 2 is distancepreserving (hence continuous) in the pseudometrics d ”