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Hyperspaces of Sets (Pure and Applied Mathematics): Sam B. Nadler

This is the first book devoted to hyperspace theory. Primarily, the book is concerned with the structure and embedding properties of the two hyperspaces 2^X and C(X) when X is a metric continuum.

I have attempted to treat much of the material from a unified point of view by using Whitney maps. These types of maps are defined and constructed in the first chapter of the book. They are used to obtain a systematic treatment of much of the theory of hyperspaces, including new proofs of some classical theorems. The idea of using Whitney maps in hyperspaces is due to Kelley [165] - I have expanded on his idea.

The book may be considered as being essentially in three parts. Chapter 0 and Chapter I contain the basic classical results about hyperspaces and, thus, they comprise the first part. Many results left around the arc structure of hyperspaces. This arc structure is studied in depth throughout Chapter I. Because the material in Chapters 0 and I is so fundamental to hyperspace theory, many detailed proofs are included. Chapters II through XVII, which cover a variety of topics about 2^X and C(X), may be considered as being the second part of the book. Chapters XVIII and XIX deal with hyperspaces other than 2^X and C(X), and thus these two chapters comprise the third part. In Chapter XVIII, spaces of compact convex sets are studied. Results and techniques in infinite-dimensional topology, functional analysis, and convexity theory are used and some applications to these areas are presented.

In Chapter XIX some other spaces of special sets are discussed. The various topics in the book are, for the most part, covered thoroughly. Of notable exception is the theory of selections. Selections are discussed from the viewpoint of selections from sub-spaces of 2 and ?(X) rather than from the viewpoint of selec- selections for set-valued mappings. This is in keeping with the emphasis of the book. The general theory and numerous applications of selections would be too lengthy to discuss here. Many references are given for the interested reader. Theorems are not always stated in the order in which they were originally proved, and sometimes results and ideas many years apart are combined. For example: See A.8) and A.25) where results and ideas of Borsuk-Mazurkiewicz and of Kelley are combined, or see the proof of A.61) where the notion of terminal continua is used. However, though the material is not always treated from an historical perspective, there are a number of historical comments throughout the book;

Another comment about the theorems in the book is appropriate. The theorems are usually stated so as to be "true to the literature." By this I mean that they are usually stated with exactly the same information as is in the referenced paper. Sometimes, because of the exposition, this necessitates a minor generalization of a particular theorem or a change in its emphasis. When generalizations or modifications are combined directly with a theorem in the literature, statements are made to that effect. Hopefully, this results in the book being a convenient and accurate reference book to the literature.

The literary style in the book is in general governed by the following factors. One rarely reads a mathematics book, especially one of this size, from beginning to end - frequently one picks a topic and opens the book to read about that topic. There is almost no standard terminology and notation in hyperspace theory, and many fundamental ideas and theorems have only recently been discovered. Thus, as an aid to the reader, the exposition often contains many references to related material, and the proofs reference even the most basic facts and definitions so that the reader can easily find them in the book. By necessity, such references are given using the numbering system employed in the book. As a consequence, smoothness of style is often sacrificed - the overriding objective being to enable the reader to easily pinpoint all the information in the book about any particular topic. With this in mind, I have also included a comprehensive subject index.

Throughout the book there are a number of questions. When the word "Question" appears (to be differentiated from the "Exercises" at the ends of Chapters 0 and I), this means that the answer to the question is not known. Thus, such questions are "research questions." Many of these questions are my own and originate with the book. The same holds true for a number of theorems, examples, etc. Also, a number of new concepts are introduced in the book such as C-determined, C*-smooth, touching set, and Whitney hyperspace. In addition, some parts of the book are devoted almost entirely to topics which are original with the book - for example, Chapter XV. Conventions for credit are in Footnotes 1 and 5 on p.52.

In summary, the book is a text; a research monograph containing many original results and unanswered questions; and a survey of and a guide to the literature.

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