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Topologies on closed and closed convex sets. (English) Zbl 0792.54008
Mathematics and its Applications (Dordrecht). 268. Dordrecht: Kluwer Academic Publishers. xi, 340 p. (1993).
Let $$X$$ be a (metric, topological or topological vector) space and let $$A$$ be a family of (closed, compact, or closed convex etc.) subsets of $$X$$. From the point of view of theory as well as applications, one would like to study topologies (called hypertopologies) or convergences on $$A$$. At the turn of this century, Painlevé describes a convergence on $$A$$ which was rediscovered by Kuratowski in the thirties. In case $$X$$ is a metric space, Pompeiu defined a metric on $$A$$ and an equivalent metric was studied by Hausdorff in the twenties. At the same time, in the case of a topological space $$X$$, Vietoris defined a hit-and-miss topology on $$A$$. The first major study of hyperspaces was made by Michael in the fifties. This subject which has applications in Optimization, Measure Theory, Mathematical Economics etc. grew at a modest rate and there are several monographs available in the literature dealing with geometrical viewpoint as well as with a view to applications. The subject of hypertopologies just exploded in the eighties and at the moment, a large number of mathematicians all over the world (Topologists, Analysts, Applied Mathematicians) are active in this area. Among them the author of this book is an acknowledged leader. Besides writing a large number of important papers on his own he has also collaborated with many of the current workers. His papers contain deep, beautiful and unexpectedly surprizing results which are well motivated and clearly exposed. Thus the author is most qualified to write on this subject and the present book is most welcome.
The monograph is written in the form of a textbook providing motivation for each concept and each result as well as providing exercises at the end of each section. Several diagrams are provided to illustrate the concepts. Much of the material, including the exercises, is taken from the recent literature and contains results discovered the day before the manuscript was sent to the publisher. Prerequisites for studying the book are basic courses in Topology and Functional Analysis that most beginning graduates study. Nevertheless the author has taken a lot of pains to make the book self-contained and he has included many basic results from the above subjects that are needed. The book includes a very comprehensive bibliography and an index which makes it quite useful for the workers in the field. At the end of the book there are notes and references on each chapter wherein the results are put in historical perspective and in some cases, there are suggestions for further research.
For the most part the monograph contains the theory of topologies defined on the closed subsets of a metric space and on closed convex subsets of a normed linear space. The central theme is the interplay among topologies, set convergence and set functionals. Several new hypertopologies and convergences have been introduced in the literature in recent times; the monograph gives an exhaustive and unifying account of these which would be of great help to beginning as well as advanced students or to workers in the field. Although the base space may be metrizable, the associated hypertopologies are not necessarily metrizable but are, in most cases, uniformizable.
In section (4.4) the author presents the results in the setting of a Hausdorff uniform space $$X$$. In the opinion of the reviewer, several other sections would benefit from this generality because in the metric setting some of the results are not clear and are harder to prove. As an example 3.3.12 states that when $$X$$ is a metrizable space, the locally finite hypertopology $$T_{\text{lf}}$$ on $$CL(X)$$ equals $$T_{\text{sup}}$$, the supremum of the Hausdorff metric topologies $$H_{\text{Hd}}$$ for each compatible metric $$d$$ on $$X$$. In fact, $$T_{\text{sup}}$$ is generated by the fine uniformity on $$X$$ via the Hausdorff uniformity and equals $$T_{\text{sup}}$$ because $$X$$ is normal! [See the reviewer and P. L. Sharma, Proc. Am. Math. Soc. 103, No. 2, 641-646 (1988; Zbl 0655.54008)]. This is not clear from the exposition in the monograph.
The reviewer is happy to note that the subject of proximity has aquired an honoured place in the monograph, and he is delighted to recommend the book wholeheartedly to students and research workers interested in hyperspaces.

##### MSC:
 54B20 Hyperspaces in general topology 52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry 54-02 Research exposition (monographs, survey articles) pertaining to general topology 54C60 Set-valued maps in general topology 54C65 Selections in general topology 54E35 Metric spaces, metrizability 54E15 Uniform structures and generalizations 54E05 Proximity structures and generalizations 46B20 Geometry and structure of normed linear spaces 46B10 Duality and reflexivity in normed linear and Banach spaces 52A07 Convex sets in topological vector spaces (aspects of convex geometry)