For a topological space X, let CL(X) be the set of all non-empty closed subset and denote with Vietoris topology by $$(CL(X), {\mathbb {V}})$$ . In this paper, we mainly discuss hyperspace when X is an infinite countable discrete space. As application, first prove that on contains copy nth power Sorgenfrey line for each $$n\in {N}}$$ Then investigate tightness equal to set-tightness X. Moreover...

We give geometric characterisations of patch and Lawson topologies in the context of predicative point-free topology using the constructive notion of located subset. We present the patch topology of a stably locally compact formal topology by a geometric theory whose models are the points of the given topology that are located, and the Lawson topology of a continuous lattice by a geometric theo...

Random abstract simplicial complex representation provides a mathematical description of wireless networks and their topology. In order to reduce the energy consumption in this type of network, we intend to reduce the number of network nodes without modifying neither the connectivity nor the coverage of the network. In this paper, we present a reduction algorithm that lower the number of points...

We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work cla...

Let X be a Hausdorff continuum (a compact connected Hausdorff space). Let 2X (respectively, Cn(X)) denote the hyperspace of nonempty closed subsets of X (respectively, nonempty closed subsets of X with at most n components), with the Vietoris topology. We prove that if X is hereditarily indecomposable, Y is a Hausdorff continuum and 2X (respectively Cn(X)) is homeomorphic to 2Y (respectively, C...

Building on the fact that descriptive frames are coalgebras for the Vietoris functor on the category of Stone spaces, we introduce and study the concept of a Vietoris bisimulation between two descriptive modal models, together with the associated notion of bisimilarity. We prove that our notion of bisimilarity, which is defined in terms of relation lifting, coincides with Kripke bisimilarity (w...

For a poset P, let σ(P) and Γ(P) respectively denote the lattice of its Scott open subsets closed ordered by inclusion, set ΣP=(P,σ(P)). In this paper, we discuss lower Vietoris topology on give some sufficient conditions to make two topologies equal. We build an adjunction between σ(Γ(P)) prove that ΣP is core-compact iff ΣΓ(P) sober, locally compact σ(Γ(P))=υ(Γ(P)) (the topology). This answer...

Abstract There are a number of (co-)homology theories on coarse spaces. Controlled operator K -theory is by far the most popular one them. Our approach geometric. We study when does Roe-algebra space restrict to subspace. Then we show cosheaf topology. A result Mayer–Vietoris exact sequence in presence cover. compute examples as an application.