In this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line R, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers Q onto itself is homeomorphic to the infinite power of Q with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundar...

We show that the Kantorovich-Rubinstein quasi-metrics dKR and dKRa of Part I extend naturally to various spaces previsions, in particular not just linear previsions (roughly, measures) I. There are natural isomorphisms between Hoare Smyth powerdomains, as used denotational semantics, discrete sublinear normalized superlinear respectively. Turning corresponding hyperspaces, namely same but equip...

We study Vietoris–Rips and Čech complexes of metric wedge sums and metric gluings. We show that the Vietoris–Rips (resp. Čech) complex of a wedge sum, equipped with a natural metric, is homotopy equivalent to the wedge sum of the Vietoris–Rips (resp. Čech) complexes. We also provide generalizations for certain metric gluings, i.e. when two metric spaces are glued together along a common isometr...

Persistent homology is a method from computational algebraic topology that can be used to study the “shape” of data. We illustrate two filtrations — the weight rank clique filtration and the Vietoris–Rips (VR) filtration — that are commonly used in persistent homology, and we apply these filtrations to a pair of data sets that are both related to the 2016 European Union “Brexit” referendum in t...

We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the Vietoris-Rips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing ...

We give a constructive localic account of the completion of quasimetric spaces. In the context of Lawvere’s approach, using enriched categories, the points of the completion are flat left modules over the quasimetric space. The completion is a triquotient surjective image of a space of Cauchy sequences and can also be embedded in a continuous dcpo, the “ball domain”. Various examples and constr...

The symbol S( X ) denotes the hyperspace of finite unions convergent sequences in a Hausdor˛ space . This hyper-space is endowed with Vietoris topology. First all, we give characterization sequence ). Then consider some cardinal invariants on ), and compare character, pseudocharacter, sn-character, so-character, network weight cs-network S ( corresponding function Moreover, rank k -diagonal 2 s...

Given a metric space X and a distance threshold r > 0, the Vietoris–Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of JeanClaude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris–Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris–Rips...

James Ax showed that, in each characteristic, there is a natural bijection from the space of complete theories of pseudo-finite fields, in first order logic, to the set of conjugacy classes of procyclic subgroups of the absolute Galois group of the prime field. I show that when the set of subgroups of a profinite group is considered to have the Vietoris (a.k.a. hyperspace, finite, exponential, ...

n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{P}_{0}(X)$-valued set multifunctions ($X$ being a linear,topological space), in order to obtain Egoroff and Lusin type theorems forset multifunctions in the Vietoris hypertopology. Some mathematicalapplications are established and several physical implications of thema...