set we have a chain of inclusions between the Rips and Čech complexes on whenever . In general metric spaces, we have that for all , implying that the Rips and Cech filtrations are 2-interleaved with respect to the interleaving distance as introduced by Chazal et al. in 2009. It is a well-known result of Niyogi, Smale, and Weinberger that given a sufficiently dense random point cloud sample of a smooth submanifold in Euclidean space, the union of balls of a certain radius recovers the homology of the object via a deformation retraction of the Čech complex. The offset filtration is also known to be stable with respect to perturbations of the underlying data set. This follows from the fact that the offset filtration can be viewed as a sublevel-set filtration with respect to the distance function of the metric space. The stability of sublevel-set filtrations can be stated as follows: Given any two real-valued functions on a topological space such that for all , the -dimensional homology modules on the sublevel-set filtrations with respect to are point-wise finite dimensional, we have where and denote the bottleneck and sup-norm distances, respectively, and denotes the -dimensional persistent homology barcode. While first stated in 2005, this sublevel stability result also follows directly from an algebraic stability property sometimes known as the "Isometry Theorem," which was proved in one direction in 2009, and the other direction in 2011. A multiparameter extension of the offset filtration defined by considering points covered by multiple balls is given by the multicover bifiltration, and has also been an object of interest in persistent homology and computational geometry. References Mathematics Computer science Topology Geometry