Gerald Beer

Topologies on closed and closed convex sets

Preface. 1. Preliminaries. 2. Weak Topologies determined by Distance Functionals. 3. The Attouch--Wets and Hausdorff Metric Topologies. 4. Gap and Excess Functionals and Weak Topologies. 5. The Fell Topology and Kuratowski--Painleve Convergence. 6. Multifunctions - the Rudiments. 7. The Attouch--Wets Topology for Convex Functions. 8. The Slice Topology for Convex Functions. Notes and References. Bibliography. Symbols and Notation. Subject Index.

Distance functionals and suprema of hyperspace topologies

Let CL(X) denote the nonempty closed subsets of a metrizable space X. We show that the Vietoris topology on CL(X) is the weakest topology on CL(X) such that A -→ d(x, A) is continuous for each x ε X and each admissible metric d. We also give a concrete presentation of the analogous weak topology for uniformly equivalent metrics, and are led to consider for an admissible metric d the weakest topology on CL(X) such that the gap functional (A, B) -→ → {d(ta, b): a ε A, b ε B} is continuous on CL(X) × CL(X).

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

On Metric Boundedness Structures

Many years ago, S.-T. Hu gave necessary and sufficient conditions for a family of subsets of a metrizable space X to be the family of bounded sets for some admissible metric for the space. In this article, we show that in any noncompact metrizable space there are uncountably many distinct metric boundedness structures. Also, given an initial metric d for X, we look carefully at the problem of characterizing those boundedness structures determined by metrics uniformly equivalent to d. Applications to hyperspaces are given. Throughout, we rely on a dual approach to the study of metric boundedness.

Uniform continuity on bounded sets and the Attouch-Wets topology

Let CL(X) be the nonempty closed subsets of a metrizable space X. If d is a compatible metric, the metrizable Attouch-Wets topology Taw (d) on CL(X) is the topology of uniform convergence of distance functionals associated with elements of CL(X) on bounded subsets of X. The main result of this paper shows that two compatible metrics d and p determine the same Attouch-Wets topologies if and only if they determine the same bounded sets and the same class of functions that are uniformly continpous on bounded sets.

Convergence of continuous linear functionals and their level sets

The origin in both X and X* will be denoted by 0 in the sequel. Corresponding to each y a X* and each c~ a R, we denote the level set {x a X : y(x) = c~} by L(y; e). A natural question to ask is this: can either strong (norm) or weak * convergence of a sequence in X* to a nonzero limit be described in terms of the convergence of the level sets of the linear functionals? We show that when X is complete, weak* convergence of a sequence {Yn: n a/g+} in X* to y 4= 0 is equivalent to the Kuratowski convergence of {L(y,; c0: n e/g + } to L(y; ~) for each real c~. If X is not complete, then weak * convergence does not ensure Kuratowski convergence of level sets. On the other hand, without completeness, strong convergence in X* means convergence of the distance functions for the level sets with respect to the (metrizable) topology of uniform convergence on bounded subsets of X. Most surprisingly, we show that if X is reflexive, then strong convergence in X* is equivalent to the Mosco convergence of level sets. 2. A notational convention. To avoid excessive double subscripting in the sequel, we find it convenient to introduce alternative notation for sequences and their limits (see, e.g., [19]). Suppose N is a cofinal (infinite) subset of the positive integers 2~ +. The notation {x.: n a N} will denote the infinite sequence

On convergence of closed sets in a metric space and distance functions

Let CL( X ) denote the nonempty closed subsets of a metric space X . We answer the following question: in which spaces X is the Kuratowski convergence of a sequence { C n } in CL( X ) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

Support and distance functionals for convex sets

We exhibit a simple topology on the closed convex subsets C of a normed linear space such that, under the usual ways of combining convex sets, C becomes a topological algebra. The topolog is the weakest one on C for which distance and support functionals are both continuous, as functions of the set argument. This topology is also the weakest topology for which and is a continuous function on. e×e In a reflexive space, convergence of asequence in C with respect to this topology is equivalent to pointwise plus Mosco convergence of the corresponding sequence of support functionals

Wijsman convergence: A survey

A net 〈Aλ〉 of nonempty closed sets in a metric space 〈X, d〉 is declaredWijsman convergent to a nonempty closed setA provided for eachx εX, we haved(x, A)=limλd(x, A). Interest in this convergence notion originates from the seminal work of R. Wijsman, who showed in finite dimensions that the conjugate map for proper lower semicontinuous convex functions preserves convergence in this sense, where functions are identified with their epigraphs. In this paper, we review the attempts over the last 25 years to produce infinite-dimensional extensions of Wijsmans theorem, and we look closely at the topology of Wijsman convergence in an arbitrary metric space as well. Special emphasis is given to the developments of the past five years, and several new limiting counterexamples are presented.

On the Fell topology

Let 2X denote the closed subsets of a Hausdorff topological space . The Fell topology τF on 2X has as a subbase all sets of the form {A ∈ 2X :A ∩V ≠ 0}, whereV is an open subset ofX, plus all sets of the form {A ∈ 2X :A ⊂W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for τF in terms of topological properties for τ. Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.