Related papers: Strongly far proximity and hyperspace topology

In this paper, we introduce the higher analogues of contiguity distance and its relations with simplicial Lusternik-Schnirelmann category and discrete topological complexity. Also we study the effects of geometric realisation and…

Satellite conjunctions involving "near misses" of space objects are becoming increasingly likely. One approach to risk analysis for them involves the computation of the collision probability, but this has been regarded as having some…

This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also…

We produce a large class of hyperbolic homology 3-spheres admitting arbitrarily many distinct tight contact structures. We also produce a sub-class admitting arbitrarily many distinct tight contact structures within the same homotopy class…

For a metrizable space $X$ of density $\kappa$, let $PM(X)$ be the space of continuous bounded pseudometrics on $X$ endowed with the uniform convergence topology. In this paper, its topology shall be classified as follows: (i) If $X$ is…

This paper expands on and refines some known and less well-known results about the finite subset spaces of a simplicial complex $X$ including their connectivity and their top homology groups. It also discusses the inclusion of the…

Let $(X,d,\mu)$ be a space of homogeneous type. In this note we study the relationship between two types of $s$-sets: relative to a distance and relative to a measure. We find a condition on a closed subset $F$ of $X$ under which we have…

Given a compact metric space $X$, we associate to it an inverse sequence of finite $T_0$ topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of $X$ that is a strong deformation retract. We provide a…

The notion of Scott distance between points and subsets in a metric space, a metric analogy of the Scott topology on an ordered set, is introduced, making a metric space into an approach space. Basic properties of Scott distance are…

In many high-impact applications, it is important to ensure the quality of output of a machine learning algorithm as well as its reliability in comparison with the complexity of the algorithm used. In this paper, we have initiated a…

We continue the study of ideal convergence for sequences $(x_n)$ with values in a topological space $X$ with respect to a family $\{F_\eta:\eta\in X\}$ of subsets of $X$ with $\eta\in F_\eta$, where each $F_\eta$ measures the allowed…

When one considers the collection $\mathcal{H}(\mathbb{R}^n)$ of all compact subsets of $\mathbb{R}^n$ and equip it with a topology, many questions can be asked about the topological space one ends up with. This is an example of a…

In this note, we prove that for a cobounded,Lipschitz path $\gamma:I\to\TT$, if the pull back bundle $\mathcal H_{\gamma}$ over $I$ is a strongly relatively hyperbolic metric space then there exists a geodesic $\xi$ in $\TT$ such that…

Let ${\rm Fin}(X)$ be the hyperspace consisting of non-empty finite subsets of a space $X$ endowed with the Vietoris topology. In this paper, we characterize a metrizable space $X$ whose hyperspace ${\rm Fin}(X)$ is homeomorphic to the…

Nearness theory comes into play in homotopy theory because the notion of closeness between points is essential in determining whether two spaces are homotopy equivalent. While nearness theory and homotopy theory have different focuses and…

This article introduces and studies the tight approximation property, a property of algebraic varieties defined over the function field of a complex or real curve that refines the weak approximation property (and the known cohomological…

In this article we study two "strong" topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space. Namely, we construct Whitney type topologies…

We explain how to relate the ideas of Carroll geometry, matrix theory on instantonic objects, and infinite boost limits of M-theory. Based on these new insights, we explore the implications for possible holographic constructions involving a…

Assume that $\mathcal{P}$ is a topological property of a space $X$, then we say that $X$ is {\it dense-$\mathcal{P}$} if each dense subset of $X$ has the property $\mathcal{P}$. In this paper, we mainly discuss dense subsets of a space $X$,…

Let $K$ be a number field, let $X$ be a smooth integral variety over $K$, and assume that there exists a finite set of finite places $S$ of $K$ such that the $S$-integral points on $X$ are dense. Then the combined conjectures of Campana and…