Hyperspace Topologies Research Articles

Properties related to first countability and countable compactness in hyperspaces: a new approach

We study properties related to first countability and countable compactness of uniformizable Δ-hit-and-miss hyperspace topologies. We show that they are proximal hyperspace topologies. We use the Smirnov compactification as a tool in our investigation.

Developable hyperspaces are metrizable

<p>Developability of hyperspace topologies (locally finite, (bounded) Vietoris, Fell, respectively) on the nonempty closed sets is characterized. Submetrizability and having a G<sub>δ</sub>-diagonal in the hyperspace setting is also discussed.</p>

Recognizing special metrics by topological properties of the ``metri''-proximal hyperspace

In this paper, we first characterize those compatible metrics $d$ on a metrizable space $X$ which give rise to a connected $d$-proximal hyperspace. We show that the space of irrational numbers, in particular, admits a complete metric with this property and, as a consequence, we get a negative answer to a question of [11] about selections for hyperspace topologies. Next, we characterize the compatible metrics on $X$ which are uniformly equivalent to ultrametrics showing that this is equivalent to the zero-dimensionality of the corresponding proximal hyperspaces. Applications and related results about other disconnectedness-like properties of proximal hyperspaces are obtained.

Completeness properties of the generalized compact-open topology on partial functions with closed domains

The primary goal of the paper is to investigate the Baire property and weak α -favorability for the generalized compact-open topology τ C on the space P of continuous partial functions f :A→Y with a closed domain A⊂X . Various sufficient and necessary conditions are given. It is shown, e.g., that ( P ,τ C ) is weakly α -favorable (and hence a Baire space), if X is a locally compact paracompact space and Y is a regular space having a completely metrizable dense subspace. As corollaries we get sufficient conditions for Baireness and weak α -favorability of the graph topology of Brandi and Ceppitelli introduced for applications in differential equations, as well as of the Fell hyperspace topology. The relationship between τ C , the compact-open and Fell topologies, respectively is studied; moreover, a topological game is introduced and studied in order to facilitate the exposition of the above results.

Selections for Vietoris-Like Hyperspace Topologies

A general selection theorem for a Vietoris-like hyperspace topology is obtained and, as a result, several known selection theorems for both ‘topology’- and ‘metric’-generated hyperspace topologies are improved from a common point of view. Applications are presented in characterizing classes of metrizable spaces by continuous selections for their hyperspaces. 1991 Mathematics Subject Classification: primary 54B20, 54C65; secondary 54E45, 54D65, 54E40, 54C10, 54G12.

Relative Compactness for Capacities, Measures, Upper Semicontinuous Functions and Closed Sets

We prove an Ascoli theorem for capacities. This theorem which characterizes relatively compact sets of capacities is widely applicable and many Ascoli theorems for particular classes of capacities can immediately be deduced as corollaries. Indeed it is usually necessary only to demonstrate that these classes are closed and then to simplify the characterization when possible. In particular, we show that the proof of the classical Prohorov theorem can be naturally factored into the shorter proof of the Ascoli theorem for capacities and into the somewhat longer proof that the class of probability measures is closed in the class of capacities. We also deduce new and known Ascoli theorems for sup measures, upper semi-continuous functions, the Vietoris hyperspace topology, and various classes of measures.

Decomposition Properties of Hyperspace Topologies

Let X be a T1 topological space and Δ a nonempty family of closed subsets of X. We study the hit-and-miss hyperspace topology generated by Δ in terms of its upper and lower parts, focusing on first and second countability and quasi-uniformization. We also obtain some new results on the Vietoris and Fell topologies.

The Wijsman and Attouch-Wets topologies on hyperspaces revisited

In this paper we will prove that, for an arbitrary metric space X and a fairly arbitrary collection Σ of subsets of X, it is possible to endow the hyperspace CL( X) of all nonempty closed subsets of X (to be identified with their distance functionals) with a canonical distance function having the topology of uniform convergence on members of Σ as topological coreflection and the Hausdorff metric as metric coreflection. For particular choices of Σ, we obtain canonical distance functions overlying the Wijsman and Attouch-Wets topologies. Consequently we apply the general theory of spaces endowed with a distance function and compare the results with those obtained for the classical hyperspace topologies. In all cases we are able to prove results which are both stronger and more general than the classical ones.

Baire spaces and hyperspace topologies

Sufficient conditions for abstract (proximal) hit-and-miss hyperspace topologies and the Wijsman hyperspace topology, respectively, are given to be Baire spaces, thus extending results of McCoy, Beer, and Costantini. Further the quasi-regularity of (proximal) hit-and-miss topologies is investigated.

On separation axioms in hyperspaces

It is shown that the well-known characterizations of separation axiomsT 2 andT 3, respectively of hit-and-miss hyperspace topologies withT 1 base space (cf. [9], [10]) are valid with no preliminary conditions on the base space.

The Infimal Value Functional and the Uniformization of Hit-and-Miss Hyperspace Topologies

The Infimal Value Functional and the Uniformization of Hit-and-Miss Hyperspace Topologies

The infimal value functional and the uniformization of hit-and-miss hyperspace topologies

We give necessary and sufficient conditions for the uniformizability of hit-and-miss and proximal hit-and-miss hyperspace topologies defined on the nonempty closed subsets CL ( X ) {\text {CL}}(X) of a Hausdorff uniform space ⟨ X , U ⟩ \langle X,\mathcal {U}\rangle . In the case of uniformizability, one can always find a family F \mathcal {F} of continuous functions on X into [0, 1] so that the hyperspace topology is the weak topology induced by { m f : f ∈ F } \{ {m_f}:f \in \mathcal {F}\} , where for each f, m f : CL ( X ) → [ 0 , 1 ] {m_f}:{\text {CL}}(X) \to [0,1] is the infimal value functional defined by m f ( A ) = inf { f ( x ) : x ∈ A } {m_f}(A) = \inf \{ f(x):x \in A\} .

Hypertopologies on function spaces

The aim of this paper is to continue Naimpally’s seminal papers [16], [17], [18], i.e. we investigate topological properties of spaces which force the coincidence of convergences of functions associated with different hyperspace topologies. For example a metric spaceX is locally compact iff the topological convergence and the convergence induced by the Fell topology coincide onC(X,IR). Moreover, the proximal topology on the space of functions, not necessarily continuous, is studied in great detail.

Weak topologies for the closed subsets of a metrizable space

The purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space. It can be shown that all of the standard hyperspace topologies ― including the Hausdorff metric topology, the Vietoris topology, the Attouch-Wets topology, the Fell topology, the locally finite topology, and the topology of Mosco convergence ― arise as weak topologies generated by families of geometric functionals defined on closed sets. A key ingredient is the simple yet beautiful interplay between topologies determined by families of gap functionals and those determined by families of Hausdorff excess functionals

Convergence of epigraphs and of sublevel sets

Let LSC(X) be the set of the proper lower semicontinuous extended real-valued functions defined on a metric spaceX. Given a sequence 〈f n〉 in LSC(X) and a functionf ∈ LSC(X), we show that convergence of 〈f n〉 tof in several variational convergence modes implies that for each α ∈ ℝ, the sublevel set at height α off is the limit, in the same variational sense, of an appropriately chosen sequence of sublevel sets of thef n, at height α n approaching α. The converse holds true whenever a form of stability of the sublevel sets of the limit function is verified. The results are obtained by regarding a hyperspace topology as the weakest topology for which each member of an appropriate family of excess functionals is upper semicontinuous, and each member of an appropriate family of gap functionals is lower semicontinuous. General facts about the representation of hyperspace topologies in this manner are given.

A FAMILY OF GRAPH TOPOLOGIES FOR FUNCTION SPACES AND THEIR ASSOCIATED WEAK GRAPH TOPOLOGIES

Abstract Let X, Y be topological spaces. We denote by Yx , C(X, Y) and G(X, Y) the set of all functions from X to Y, the set of all continuous functions and of all functions with closed graphs respectively. Now as is well-known, identifying a function f with its graph Γfwe can transfer a (known) hyperspace topology for the product space X x Y to the function space Yx (or to a subspace of Yx ). A function space topology which is generated in this way we will call a graph topology. For a broad and important family of such graph topologies for each member of this family we can define an associated weaker graph topology. This family will be characterized, and we find general conditions implying the joint continuity and the validity of the HausdorfF separation axiom for the members of this family and their associated weak graph topologies. Applying these general assertions we get back corresponding results for the weak Fell topology and the Fell topology respectively and we get new results for graph topologies...

Uniform properties and hyperspace topologies for ℵ-uniformities

Let X be a completely regular space and ℵ an infinite cardinal number. The ℵ-uniformity of X generated by all open normal coverings of X with cardinality ⩽ℵ, is the weakest one with the following property: any continuous function from X to any metric space of weight⩽ℵ is uniformly continuous. Any continuous function from a uniform space X to any metric space of weight⩽ℵ is uniformly continuous iff any locally finite covering of cozero-sets of cardinality ⩽ℵ is uniform. With ℵ-collectionwise normality, any continuous function from X to any metric space of weight ⩽ℵ and uniform dimension ⩽1 is uniformly continuous iff any discrete family of subsets of X with cardinality ⩽ℵ is uniformly discrete. The uniform hypertopologies induced via the Hausdorff uniformity on the hyperspace 2 x of X from the ℵ 1-uniformity, generated by the family of all continuous functions from X to any metric space of density ⩽ℵ and uniform dimension ⩽1 and from the ℵ-uniformity agree. Further, both agree with a Vietoris-type topology iff X is normal.

Fine Uniformity and the Locally Finite Hyperspace Topology

It is shown that a uniformizable space $X$ is normal iff the locally finite topology ${e^\tau }$ on the hyperspace ${2^X}$ coincides with the topology transmitted by the fine uniformity of $X$. We also prove that, for $X$ normal, the topology ${e^\tau }$ is first countable only if the set of limit points $X’$ of $X$ is countably compact. Applications of these results to pseudocompactness and Atsuji spaces are given.

Fine uniformity and the locally finite hyperspace topology

It is shown that a uniformizable space X X is normal iff the locally finite topology e τ {e^\tau } on the hyperspace 2 X {2^X} coincides with the topology transmitted by the fine uniformity of X X . We also prove that, for X X normal, the topology e τ {e^\tau } is first countable only if the set of limit points X ′ X’ of X X is countably compact. Applications of these results to pseudocompactness and Atsuji spaces are given.

Topologies of the Hyperspace which Are Induced by Extension Spaces

AbstractThe hyperspace of all nonvoid closed subsets of a topological space will be topologized by means of different methods; each of them generalizes known definitions of special hyperspace topologies. One method uses the VIETORIS topology in the hyperspaces of certain extension spaces. Another one uses certain systems of closed subsets and leads to just the same class of hyperspace topologies. In natural order VIETORIS topology is the supremum of this class, and for locally compact spaces FLACHSMEYER topology is the infimum of this class.