Moy, Michael, authorAdams, Henry, advisorKing, Emily, committee memberBen-Hur, Asa, committee member2021-06-072021-06-072021https://hdl.handle.net/10217/232524Persistent homology often begins with a filtered simplicial complex, such as the Vietoris–Ripscomplex or the Čech complex, the vertex set of which is a metric space. An important result, the stability of persistent homology, shows that for certain types of filtered simplicial complexes, two metric spaces that are close in the Gromov–Hausdorff distance result in persistence diagrams that are close in the bottleneck distance. The recent interest in persistent homology has motivated work to better understand the homotopy types and persistent homology of these commonly used simplicial complexes. This has led to the definition of metric thickenings, which agree with simplicial complexes for finite vertex sets but may have different topologies for infinite vertex sets. We prove Vietoris–Rips metric thickenings and Čech metric thickenings have the same persistence diagrams as their corresponding simplicial complexes for all totally bounded metric spaces. This immediately implies the stability of persistent homology for these metric thickenings.born digitalmasters thesesengCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.metric thickeningspersistent homologyapplied topologytopological data analysispersistence stabilityText