The Vietoris-Rips complex characterizes the topology of a point set. This complex is popular in topological data analysis as its construction extends easily to higher dimensions. We formulate a two-phase approach for its construction that separates geometry from topology. We survey methods for the first phase, give three algorithms for the second phase, implement all algorithms, and present exp...

Given a generalized continuum $X$, let $\operatorname{CL}_{{\rm F}}(X)$ and $\operatorname{C}(X)$ denote its hyperspaces of (non-empty) closed subsets subcontinua, respectively, with the Fell topology ($=$ Vietoris on $\operatorname{C}(X)

For a regular space X, the hyperspace (CL(X),τF) (resp., (CL(X),τV)) is of all nonempty closed subsets X with Fell topology Vietoris topology). In this paper, we give characterization such that countable character subsets. We mainly prove has on each subset if and only compact metrizable, locally separable metrizable. Moreover, (K(X),τV) have compact-Gδ-property every

In this paper, for a given sequentially Yoneda-complete T1 quasi-metric space (X, d), the domain theoretic models of the hyperspace K0(X) of nonempty compact subsets of (X, d) are studied. To this end, the ω-Plotkin domain of the space of formal balls BX, denoted by CBX is considered. This domain is given as the chain completion of the set of all finite subsets of BX with respect to the Egli-Mi...

If a sequence of random closed sets Xn in a separable complete metric space converges in distribution in the Wijsman topology to X, then the corresponding sequence of cores (sets of probability measures dominated by the capacity functional of Xn) converges to the core of the capacity of X. Core convergence is achieved not only in the Wijsman topology, but even in the stronger Vietoris topology....

Fix a finite set of points in Euclidean n-space En, thought of as a point-cloud sampling of a certain domain D ⊂ En. The VietorisRips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Vietoris-Rips complex to En that has ...

We show that filtering the barycentric decomposition of a Čech complex by the cardinality of the vertices captures precisely the topology of k-covered regions among a collection of balls for all values of k. Moreover, we relate this result to the Vietoris-Rips complex to get an approximation in terms of the persistent homology.

We prove that the space of causal curves between compact subsets of a separable globally hyperbolic poset is itself compact in the Vietoris topology. Although this result implies the usual result in general relativity, its proof does not require the use of geometry or differentiable structure.