On C^0-persistent homology and trees
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The study of the topology of the superlevel sets of stochastic processes on [0,1] in probability led to the introduction of -trees which characterize the connected components of the superlevel-sets. We provide a generalization of this construction to more general deterministic continuous functions on some path-connected, compact topological set X and reconcile the probabilistic approach with the traditional methods of persistent homology. We provide an algorithm which functorially links the tree T_f associated to a function f and study some invariants of these trees, which in 1D turn out to be linked to the regularity of f.
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