Hyperspace Topologies Research Articles

Cofinal completeness vis-á-vis hyperspaces

A metric space is called cofinally complete if every cofinally Cauchy sequence in it has a cluster point. For a metric space (X, d), we consider the set AC(X) of almost nowhere locally compact sets (Beer and Di Maio in Acta Math Hung 134(3):322–343, 2012), which is a subset of the set of closed subsets in X. We consider some hyperspace topologies on the set AC(X) which it inherits from the set of non-empty closed subsets in X as subspace topologies and characterize cofinally complete metric spaces in terms of relations to Hausdorff metric topology, proximal topology, Vietoris topology and locally finite topology on AC(X). We characterize cofinally complete metric spaces in terms of some function space topologies as well. Furthermore, we characterize the class of metric spaces for which the corresponding space AC(X) equipped with the Hausdorff metric topology is cofinally complete.

Weak convergence of probability measures to Choquet capacity functionals

In the definition of weak convergence of probability measures it is assumed that the limit is a probability measure as well. We get rid of this assumption and require that the limit merely needs to be a Choquet-capacity functional. In terms of random variables this means that the distributional limit no longer is a random point, but a random closed set, namely one uniquely determined by the Choquet capacity. For our extended notion of weak convergence there is a counterpart of the portmanteau theorem. Moreover, we demonstrate basic relations to the theory of random closed sets with emphasis on weak convergence in hyperspace topologies including two correspondence theorems. Finally, the approach carries over to sequences of Choquet capacities.

Open quotients of trivial vector bundles

Given an arbitrary topological complex vector space A, a quotient vector bundle for A is a quotient of a trivial vector bundle π2:A×X→X by a fiberwise linear continuous open surjection. We show that this notion subsumes that of a Banach bundle over a locally compact Hausdorff space X. Hyperspaces consisting of linear subspaces of A, topologized with natural topologies that include the lower Vietoris topology and the Fell topology, provide classifying spaces for various classes of quotient vector bundles, in a way that generalizes the classification of locally trivial vector bundles by Grassmannians. If A is normed, a finer hyperspace topology is introduced that classifies bundles with continuous norm, including Banach bundles, and such that bundles of constant finite rank must be locally trivial.

Baire spaces and hyperspace topologies revisited

It is the purpose of this paper to show how to use approach spaces to get a unified method of proving Baireness of various hyperspace topologies. This generalizes results spread in the literature including the general (proximal) hit-and-miss topologies, as well as various topologies generated by gap and excess functionals. It is also shown that the Vietoris hyperspace can be non-Baire even if the base space is a 2nd countable Hausdorff Baire space.

On First Countability of a New Hyperspace Topology

AbstractIn this paper the notion of first countability of the hyperspace (θ (X), τ ) has been studied and being compared with that of X.

Bornological modifications of hyperspace topologies

The bornological convergence structures that have been studied recently as generalizations of Attouch–Wets convergence define pretopologies on hyperspaces. In this paper we characterize the topological reflections of these pretopologies and translate the constructions necessary to define bornological convergence to a broader spectrum of hyperspace topologies.

Fell continuous multi-sections

The paper is devoted to several characterisations of metacompact-like properties in terms of special sections continuous with respect to the Fell hyperspace topology. These results fit naturally within the framework of covering properties described by similar sections, but continuous with respect to other hyperspace topologies.

On normality of the Wijsman topology

Let (X, ρ) be a metric space and (CL(X), W ρ ) be the hyperspace of all nonempty closed subsets of X equipped with the Wijsman topology. The Wijsman topology is one of the most important classical hyperspace topologies. We give a partial answer to a question posed in Di Maio (Quaderni di Matematica, 3:55–92, 1998) whether the normality of (CL(X), W ρ ) is equivalent to its metrizability. If (X, ρ) is a linear metric space, then (CL(X), W ρ ) is normal if and only if (CL(X), W ρ ) is metrizable. Some further results concerning normality of the Wijsman topology on CL(X) are also proved.

Topological dynamics on hyperspaces

In this paper we wish to relate the dynamics of the base map to the dynamics of the induced map. In the process, we obtain conditions on the endowed hyperspace topology under which the chaotic behaviour of the map on the base space is inherited by the induced map on the hyperspace. Several of the known results come up as corollaries to our results. We also discuss some metric related dynamical properties on the hyperspace that cannot be deduced for the base dynamics.

Selection principles in hyperspaces with generalized Vietoris topologies

We continue investigations of Čech closure spaces and their hyperspaces started in [M. Mršević, M. Jelić, Selection principles and hyperspace topologies in closure spaces, J. Korean Math. Soc. 43 (2006) 1099–1114] and [M. Mršević, M. Jelić, Selection principles, γ-sets and α i -properties in Čech closure spaces, Topology Appl., in press], focusing on generalized upper and lower Vietoris topologies.

Atsuji completions vis-à-vis hyperspaces

Abstract A metric space (X, d) is called an Atsuji space if every real-valued continuous function on (X, d) is uniformly continuous. It is well-known that an Atsuji space must be complete. A metric space (X, d) is said to have an Atsuji completion if its completion ($$ \hat X $$, d) is an Atsuji space. In this paper, we study twelve equivalent (external) characterizations for a metric space to have an Atsuji completion in terms of hyperspace topologies. We also characterize topologically those metrizable spaces whose completions are Atsuji spaces.

Selection principles, γ-sets and [formula omitted]-properties in Čech closure spaces

We continue investigations of Čech closure spaces and their hyperspaces started in [M. Mršević, M. Jelić, Selection principles and hyperspace topologies in closure spaces, J. Korean Math. Soc. 43 (2006) 1099–1114], focusing on sets related to γ-sets and selection principles related to α i -properties.

Symmetric Bombay topology

The subject of hyperspace topologies on closed or closed and compact subsets of a topological space X began in the early part of the last century with the discoveries of Hausdorff metric and Vietoris hit-and-miss topology. In course of time, several hyperspace topologies were discovered either for solving some problems in Applied or Pure Mathematics or as natural generalizations of the existing ones. Each hyperspace topology can be split into a lower and an upper part. In the upper part the original set inclusion of Vietoris was generalized to proximal set inclusion. Then the topologization of the Wijsman topology led to the upper Bombay topology which involves two proximities. In all these developments the lower topology, involving intersection of finitely many open sets, was generalized to locally finite families but intersection was left unchanged. Recently the authors studied symmetric proximal topology in which proximity was used for the first time in the lower part replacing intersection with its generalization: nearness. In this paper we use two proximities also in the lower part and we obtain the lower Bombay hypertopology. Consequently, a new hypertopology arises in a natural way: the symmetric Bombay topology which is the join of a lower and an upper Bombay topology.

On metrization of the hit-or-miss topology using Alexandroff compactification

When the hit-or-miss topology is employed, the space of all closed subsets of a Hausdorff, locally compact and second countable space (HLCSC) is known to be Hausdorff, compact and second countable, thus metrizable. This paper investigates metrics on this space by using Alexandroff compactification technique, with a more general metrization procedure developed. A note concerning the necessity of the condition HLCSC on E is included. With the constructed metric, we investigate a hyperspace Birkhoff ergodic theorem to explore the connection between orbital behaviors of hyperspace dynamical systems and Choquet capacities of random closed sets. Moreover, relations between the hit-or-miss topology and other hyperspace topologies or metrics such as the Vietoris topology, Hausdorff metric and Hausdorff–Buseman metric are also given.

On hyperspace topologies via distance functionals in quasi-metric spaces

Let (X,d) be a quasi-pseudo-metric space. We investigate hyperspace topologies on P0(X) defined by distance functionals. In particular, the K-topology is introduced and compared with other hyperspace topologies. Some properties of the Wijsman topology and the K-topology are explored.

Topologies on uniform hyperspaces

This paper includes the proofs of results announced in [3], as well as other results deriving from the Isbell-Smith-Ward problem of comparability of Hausdorff uniform topologies on hyperspaces of uniform spaces. In particular we give (in Theorem 6) simple conditions on a uniform space sufficient for H-singularity (no other uniformity induces the same hyperspace topology). The notion of association map is introduced, and properties of a map between uniform spaces are related to properties of the induced hyperspace map, thus generalizing and unifying results of F. Albrecht, D. Hammond Smith and V.Z. Poljakov, and establishing various conditions sufficient for uniform continuity.

The Reznichenko property and the Pytkeev property in hyperspaces

We investigate two closure-type properties, the Reznichenko property and the Pytkeev property, in hyperspace topologies.

Fell continuous selections and topologically well‐orderable spaces

The present paper extends the idea of characterizing topological properties of a space X by means of continuous selections for its closed subsets (X) endowed with a “natural” hyperspace topology. In this particular case, it is proved that the property of X to be topologically well-orderable is equivalent to the existence of a selection for (X) which is continuous with respect to the Fell topology.

Tightness, character and related properties of hyperspace topologies

Given a Hausdorff space X, we calculate the tightness and the character of the hyperspace CL ∅( X) of X, endowed with either the co-compact or the lower Vietoris topology, and give some estimates for the tightness of CL ∅( X), endowed with the Fell topology. Some properties related to first-countability and countable tightness, such as sequentiality, Fréchet property and, less directly, radiality and pseudoradiality, are investigated as well. To carry out our investigation, we also consider on the base space X several cardinal functions, and we compare some of them (which are newly defined or not so well known) with other classical ones, obtaining results and counterexamples which may be of some independent interest.

Some problems on selections for hyperspace topologies

The theory of hyperspaces has attracted the attention of many mathematicians who have found a large variety of its applications during the last decades. The theory has taken also its natural course and has yielded lots of problems which, besides their independent inner beauty, provide ties with numerous classical fields of mathematics. In the present note we are concerned with some open problems about selections for hyperspace topologies which have been in the scope of our recent research interests.