Notes and References Chapter 1 The material in § 1.1 is very well-known. There are many accessible accounts of general topology; for our purposes, we recommend the monographs of Dugundji [Dug] and Willard [Wil]. Somewhat more imposing is the encyclopedic monograph of [En], which gives an unmatched account of the literature. The monograph of Berge [Bg], although omitting many aspects of general topology, presents many of the constructs of set-valued analysis that the reader will encounter in subsequent chapters, and at the same time presents significant material on convex sets and the elements of nonlinear analysis. In some sense, our text seeks to maintain the flavor of [Bg], often walking the border between point-set topology and geometric functional analysis. For the latter subject, we recommend the monographs of Holmes [Holl-2] and Giles [Gil]. The ball measure of noncompactness as introduced in § 1.1 and its relatives are considered in the monograph of Banas and Goebel [BaG]. Kuratowski's Theorem [Ku1] presented in Exercise 1.1.4 has numerous applications; it is a curious fact that for a decreasing sequence <An> of closed convex subsets of a Banach space, limn--+ oo X(An\An+l) = 0 is enough to guarantee nonempty intersection, as shown by Cellina [CeI4]. One of the most striking aspects of the theory of topologies on closed sets is that the most important of them have attractive presentations as weak: topologies determined by geometric set functionals. Although weak: topologies are standard tools in modem linear analysis, it is difficult to find facts about them collected in one place. For this reason § 1.2 exists. We mention that the idea of weak: topology as it is used in functional analysis originates with von Neumann [Neu]. In this book we only consider uniformities as described by entourages; altematively, they may be approached through uniform covers v [ls] or through pseudo-metrics [Dug]. For accounts of Baire spaces and Cech complete spaces, the reader may consult [En] and Haworth and McCoy [HaM]. The notion of semicontinuous function originates with Baire [Bai]. Lower semicontinous functions rather than upper semicontinuous functions are usually considered primitive in one-sided analysis, as the theory has its origins in the study of convex rather than concave functions. The systematic construction of proofs for results involving semicontinuous functions based on the manipulation of epigraphs, rather than treating such functions as transformations, owes a great deal to the monograph of Rockafellar [Roc4]. Theorem 1.3.7 for real-valued functions defined on a metric space is due to Hahn [Hah!]; a Tl space admits such continuous approximations if and only if it is perfectly normal [Ton]. There exists a literature on semicontinuous functions with values in a partially ordered topological space, where there is some compatibility between the order and the topology (see, e.g., [PT,Be6,How,McC2]). The Separation Theorem is the corners tone of geometric functional analysis, and it is of course possible to formulate it much more generally, even in a linear space without topological structure. It may be used to prove the general Hahn-Banach Theorem, or it may be derived from this extension result [HoI2,Gil]. Modem convex function theory in a sense originated with the lecture notes of Fenchel [Fen], and reached its maturity much through the efforts ofRockafellar and Moreau in the 1960's [Roc4,Mor2]. The classical approach to the subject, with a particularly attractive treatment of convex functions and inequalities, is documented in [RV]. There is an enormous literature on convex cones and their application; the reader may consult Aubin and Frankowska [AuF]. Purely geometrical aspects of the theory of convex sets in locally convex spaces are considered in the historically important volume of Valentine [Val]. For an introduction to the theory of 306 NOTES AND REFERENCES 307 convex sets in finite dimensions, we recommend [Lay,MS]. The differentiation of convex functions has been a focal point of research in recent years (see, e.g., [Gil,Ph]). The idea of excess (ecart) originates with Pompieu in 1905 [Porn]. The name "gap functional" was forcefully suggested to the author by Penot. Lemma 1.5.1 can be found in [Cnt], although it was known much earlier for bounded sets. Chapter 2 In view of the centrality of weak topologies in modern functional analysis, it is a little odd that the the creation of a theory of hyperspaces based on weak topologies was not given adecent try until weIl after 1980. Efforts in this direction have their roots in the study of topologies on lower semicontinuous functions, both convex and nonconvex [Verl-2,Jol], but it was the 1985 paper of Francaviglia, Levi and Lechicki [FLL], who considered distance functionals with a fixed point argument as a function of a set variable, that focused attention on the general method. The notion of strict d-inclusion originates with Costantini, Levi and Zierninska [CLZ]. Paralleling Theorem 2.1.10, necessary and sufficient conditions for Wijsman topologies to agree for closed convex sets have been identified by Corradini [Cor2]. The Vietoris topology was introduced in the early 1920's by Vietoris in [Viel-2J, and its basic topological properties were identified by Michael [Mic 1] in a 1951 paper that set the agenda for the study of hyperspaces for the next twenty years. In terms of later developments, we cite Keesling's deep study of compactness in the hyperspace [Keel-2J, as weIl as Smithson's interesting results on first countability [Sm4]. The term "proximal hit-and-miss topology" originates with Naimpally and his collaborators [BLLN, DCNS], although the proximal topology of §2.2 is implicit in the work of Nachmann [Nac]. Sequential convergence with respect to the proximal topology was studied by Fisher [Fis] and is sometimes called Fisher convergence in the literature (see, e.g., [BaP]). A number of authors had tried to pin down the precise relationship between the Wijsman topology, the ball proximal topology, and the topology of Exercise 2.2.12 (see, e.g., [FLL,Be8, DMN,BTa1] before Hola and Lucchetti [HL] clarified the issue, using the constructs of [CLZ]. Hit-and-miss topologies in the abstract were studied by Poppe [Popl-2] in papers that should have attracted more attention than they received. Given a farnily of weak topologies on a set, the supremum of these topologies is again a weak topology, as determined by the amalgamation of the defining functionals. In particular, suprema of hyperspace topologies can often be attractively expressed as weak topologies. Clean presentations of infima of hyperspace topologies are generally difficult to produce. One problem is a lack of distributivity within the lattice of hyperspace topologies. With distributivity, the infimum of the Vietoris topology and the Hausdorff metric topology ought to be the proximal topology. But as Levi, Lucchetti, and Pelant have shown, this occurs if and only if the underlying metric space is either UC or totally bounded [LLP]. UC spaces perhaps originated with Nagata [Nag] and are periodically rediscovered every few years. It is of course possible to consider UC spaces in the context of uniform spaces [Ats2,NS]. Theorem 2.3.4 is a variant of a result of the author [Be4], who used the Hausdorff metric topology in lieu of the proximal topology. For a general discussion of hyperspaces and function spaces, consult [Nai3]. The slice topology was anticipated by a hit-and-miss topology compatible with Mosco convergence of sequences of convex sets [Be 10-11]. The topology itself was first introduced by Sonntag and Zalinescu [SZ2] as a weak topology determined by a farnily of gap functionals (see Corollary 4.l.7) as part of an overall program to classify hyperspace topologies. Around the same time, the author [Be19-20] introduced the slice topology in 308 NOTES AND REFERENCES much the same way in an attempt to come up with a topology a good deal stronger than the Wijsman topology that remained compatible with the measurability of convex-valued multifunctions (see §6.5). For lower semicontinuous convex functions identified with their epigraphs, the topology agrees with a topology of Joly [Jol] defined in terms of the lower semicontinuity of epigraphieal multifunctions, as we show in §8.2 The author does not know whether the Wijsman topology corresponding to an arbitrary compatible metrie for a Polish space is necessarily Polish. This is true in the special case that the metrie d is totally bounded, as established by Effros [Eff]. The author is hopeful that more transparent proofs of the results of §2.5 can be found, as weH as a useful complete metrie for the Wijsman topology for the closed subsets of a Polish space. Chapter 3 Attouch-Wets convergence appears in Mosco's initial 1969 paper on Mosco convergence [Mos1], but was not studied in depth until many years later by Attouch and Wets [AW1-2]. Theorem 3.1.3 was first proved by Attouch, Lucchetti, and Wets [ALW], modifying the standard proof of completeness for the Hausdorff metric topology, as established for bounded sets by Hahn [Hah2]. The Hausdorff metric topology originates with Pompieu [Pom] , but not the Hausdorff metrie itself, as Pompieu worked with the equivalent metric p(A, B) =ecJCA,B) + ed(B,A). Without question, the Hausdorff metrie topology is the most weH-studied hyperspace topology, and §3.2 merely scratches the surface. A great deal of attention has been focused on the Hausdorff metric applied to metric continua [Nad]. A central result in this theory is the Curtis-Schiori Theorem [CS]: the space of closed subsets of a locally connected metrie continuum is homeomorphic to the Hilbert cube.