From TopCom, Volume 7, #1
Handbook of the History of General Topology, Volume 2The series of Handbooks of the History of General Topology, of which this is the second volume and two more are promised, was originated after a session organized by the American Mathematical Society in San Antonio, Texas, in 1993. The project developed in a four volumes series, of 300-400 pages each, the first one being published in 1997, by the same publisher.
edited by C. E. Aull and R. Lowen
408 pp. ISBN 0-7923-5030-8
Kluwer Academic Publishers, Dordrecht, 1998
General Topology is so vast and has so many different ramifications and interactions with other fields of mathematics that would be very difficult to organize the volumes by clustering subjects or distinct directions. The organizers opted by giving a good space to the comprehensive work of the main proponents of the field that emerged in the beginning of the 20th century. And considerable space is given to sub-areas which developed in the course of the century. The subjects covered are wide-ranging and eclectic.
In the very beginning of the Introduction, the editors state that "it was felt that it was important to undertake the project while topologists who knew some of the early researchers were still active." Indeed, the authors of the articles were chosen among contributors, in some cases the introducers of fields. It is significant that the first volume opens with Felix Hausdorff, whose seminal book Grundzüge der Mengenlehre (1914) is a landmark in the field. Several other contributors are studied and different specific sub-fields are analyzed. This helps to situate the volume under review.
As the previous one, this second volume analyses personalities and specific topics. The geographical spread of the contributors reveal the predominance of Europeans and North Americans in the development of the General Topology. Regrettably, the book does not have a section on the profile of the contributors, such as "About the Contributors", which is common in books of this kind by this publisher. There is a possibility to repair this omission preparing a comprehensive chapter on the contributors in the next volumes. Also, the volume would benefit from an expanded Index. Since page numbering is integrated [this volume begins with page 399], we can expected an overall index in Volume 4.
Three articles focus on the Warsaw School of Mathematics. The individual contributions and influences of Waclaw Sierspinski (1882-1969) and of Kazimierz Kuratowski (1896-1980), are discussed by Ryszard Engelking, and R. Pol discusses Stefan Mazurkiewicz (1888-1945). The individual contributions of each one are discussed, with ample bibliographical references. The three may be considered, together with Zygmund Janiszewski (1888-1920), the founding fathers of the brilliant Polish school of General Topology. It is to be hoped that a special article on Janiszewski appears in the next two volumes. They, together, were responsible for Fundamenta Mathematica, founded in 1920, a journal which, on the interwar period, exerted great influence on research in topology, set theory and logic. Together, the articles may be regarded as a history of the Warsaw School of Mathematics. Engelking and Pol, both in the same Institute, acknowledge mutual comments and this gives unity to the three opening articles. The Warsaw School of Mathematics was an idea of Sierpinski, Janiszewski and Mazurkiewicz, when they met in the Warsaw University and decided to concentrate mathematical research on set theory and its various applications, which were new domains and it would be easier to catch up with world mathematics. Later on, the younger Kuratowski joined the group and in 1932, by his initiative, the series Monografie Matematyczne was launched. The strategy adopted by these young Polish mathematicians is still regarded as an exemplary decision for emergent research groups.
The next article moves to the United States and focuses on the rare personality of R.H. Bing. The author, Michael Starbird, opted to write an article diverting from the strictly mathematical and intellectual aspects of the history of general topology and giving a sense of the personal life and style of Bing. He properly titled the article R. H. Bing's Human and Mathematical Vitality. Bing was one of the most distinguished representatives of the R. L.Moore school. As the Warsaw school was responsible for the emergence of brilliant topologists, R. L. Moore was responsible for the formation of an illustrious roll of American topologists. The Moore School was the object of an article in the first volume of this series. The career of R. H. Bing reflects the many facets of his personality. From a teacher at the Palestine High School, in Texas, where he was responsible for coaching the football and track teams, teaching mathematics classes and a variety of other classes, to be the Mildred Caldwell and Blaine Perkins Kerr Centennial Professor of Mathematics at The University of Texas at Austin, Bing had 35 Ph.D. students and authored 116 papers. Bing's fundamental contributions to point set topology is amply discussed in the paper by S. D. Shore.
There are 9 articles covering specialized subjects: From Developments to Developable Spaces, by S. D. Shore; A History of Generalized Metrizable Spaces, by R. E. Hodel; The Historical Developments of Uniform, Proximal and Nearness Concepts in Topology, by H. L. Bentley, H. Herrlich and M.Husek;Hausdorff Compactifications: A Retrospective, by R. E. Chandler and G. Faulkner; Minimal Hausdorff Spaces - Then and Now, by J. R. Porter and R. M. Stephenson Jr.; A History of Results on Orderability and Suborderability, by S. Purish; History of Continuum Theory, by J. J. Charatonik; The Alexandroff-Sorgenfrey Line, by D. E. Cameron. All the authors, with 3 exceptions, are from U.S. institutions.
Without going through each essay, all of them noteworthy, let me point out some of those which I considered most engaging. The papers by Shore and by Charatonik are longer than the others, with, respectively, 73 and 83 pages. Shore begins with the recognition of the pioneer contribution of Eliakim Hastings Moore by introducing, in 1910, the idea of development for an abstract class. This would be highly influential in the emerging concept of neighborhood and of related topological concepts, particularly the metrization problem. More recent advances in this problem are discussed in the article by R. E. Hodel. Shore recalls the number of monographs which appeared in the early 1930's, which is an indication that general topology was established as a field of mathematics research. He also mentions two papers which he considers very significant steps for the concept of developments. Both papers were published in 1937 in the Bulletin of the American Mathematical Society, both dealing with the metrization problem, respectively by F.Burton Jones and by Aline H. Frink. Then, in 1951, R. H. Bing publishes his fundamental paper on the problem, where the concept of developable space is established. Shore clearly shows the important contribution of the American school of point set topology, recognizing the fundamental role of R. H. Bing. He finishes his very interesting paper with a reference to John L.Kelley's book, General Topology, which, in his saying, became the standard text in the United States. I would say not only in the United States.
The paper by Janusz J. Charatonik, on the History of Continuum Theory, focus a different strand in General Topology, which derives of studying properties of curves and surfaces, more generally of the geometry of the Euclidean n-space. The concept of a Jordan curve, named after Camille Jordan, who stated, in 1887, that a simple closed curve in the plane cuts the plane into two regions and is their common boundary, captured the imagination of most mathematicians in the late 19th and early 20th centuries. A number of creative and pathological examples were responsible for important developments in Analysis, such as Calculus of Variations, Stability of Differential Equations, Functional Analysis, Measure and Integration, and more recent fields, such as Dynamical System and Geometric Measure Theory. Charatonik does a remarkable job in synthesizing these developments in this chapter. He presents a very generous bibliography of 729 titles. The chapter serves as an useful guide for graduate students willing to do historical research. The chapter by S. Purish, entitled A history of results on orderability and suborderability, raises some questions related to those treated by Charatonik.
The paper by H. L. Bentley, H. Herrlich and M. Husek, entitled The Historical Developments of Uniform, Proximal and Nearness Concepts in Topology, offers a good discussion on the concept of uniformity. They start relating the uneasiness with the basic concepts of Analysis during the 19th century, which were to be settled only after the introduction of concepts of uniformity, which culminated with the seminal work of André Weil, in 1937. A bibliography of almost 400 references helps the reader who wants to move into this vast field. This chapter has a peculiarity. Each section is captioned by a quote from literature. Indeed, very well selected and pertinent to the theme. This gives to mathematics a distinctive humanistic tone.
Overall, the stated goal of this Handbook is best understood if we read the three pages paper which Douglas E. Cameron wrote as a sort of preamble, while not stated as so, to his interesting paper on the Alexandroff-Sorgenfrey line. In this short paper, Cameron recognizes that his interest in the history of mathematics came while his doctoral advisor, Charles E. Aull, enticed him to look into the human facet of mathematicians. His personal recall of meeting Pavel S. Alexandroff is delightful. When Cameron thanks his mathematical father for making the mathematicians cited as real as he can, by talking about their family lives and adding anecdotes about their personal and professional lives, he gives a very good reason to do history of mathematics. Cameron translates this by advising teachers to reveal that mathematicians are humans, claiming that this "will make the learning a bit more tolerable and much more interesting". The overall tone of the book adheres to this standpoint.
This book is a valuable companion to mathematicians and historians of mathematics, as well as to teachers of mathematics both at the undergraduate and graduate levels. Surely, it gives flavor to the field. The chapters are all very well written, by recognized contributors to the field. The book is well edited, with sufficient attention to grammar and stylistic conventions.
Date: February 8, 2002