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Non-Hausdorff topology and domain theory: selected topics in point-set topology (New Mathematical Monographs 22) BY Jean Goubault-Larrecq

Non-Hausdorff topology and domain theory: selected topics in point-set topology (New Mathematical... Many pure mathematicians (indeed, many topologists ) would be surprised to learn just how much point-set topology is used these days by theoretical computer scientists. The trend began in 1972 with the introduction by Dana Scott of the topology that bears his name, initially on a continuous lattice but ultimately on an arbitrary directed-complete poset, and the nearly 500 pages of this volume bear eloquent testimony to the extent of the development that has followed. However, one feature of the topologies employed in computer science is that many of them are non-Hausdorff, which means that traditional textbooks on topology, with their emphasis on Hausdorff spaces, are less than helpful to computer science students. It is to counter this bias that the present book has been written. (However, the author overstates his case when he says on p. 1, and again on p. 49 when he introduces the Hausdorff axiom, that ‘Most traditional topology textbooks assume the Hausdorff separation condition from the very start’; in a quick survey of the textbooks on my shelves, I could not find one that does this, and many introduce the axiom substantially later than Goubault-Larrecq does. Also, despite the title, there are Hausdorff spaces in the book: indeed, after a brief summary of set theory, the first substantial chapter is devoted to metric spaces.) Nevertheless, the emphasis throughout this book is on topics that receive only fleeting attention, if any, in ‘mainstream’ topology texts: compactly generated spaces (in the non-Hausdorff sense) and cartesian closedness, quasi-metric spaces and their various notions of completeness, sober spaces and frames, stable local compactness, powerdomains and so on. Although, as already indicated, the motivation for studying this material is applications in computer science, one does not need to know any computer science in order to read the book; some applications to domain theory appear towards the end of it, but they are treated from a purely mathematical viewpoint. The reference to cartesian closedness above indicates that categorical ideas play a larger rôle in the book than they do in many texts on topology. However, one thing that I personally found irritating was that, having collected all the material he needs from set theory and ordered-set theory in a preliminary chapter, Goubault-Larrecq introduces the categorical ideas in dribs and drabs, as and when he needs them; surely it would have been neater to put this material at the beginning, too? And I would not recommend this book to anyone wishing to learn about frame theory; what it says is too little and too late in the development to be helpful. The author has a disconcerting habit of naming almost every result of significance after its supposed author(s). I was in particular intrigued to discover the identity of the ‘Johnstone space’ (p. 143) and of ‘Johnstone's theorem’ (p. 427): both are in fact due to me, but I would not count them among my most profound achievements. And, in general, the assignment of credit for ideas is distinctly haphazard: references tend to be to textbooks rather than original papers (it is significant that Scott's path-breaking 1972 paper, mentioned earlier, is not in the bibliography), with the result that the textbook authors tend to get credit for other people's ideas. A good example occurs in the very first substantial result proved in the book, the Bourbaki–Witt fixed-point theorem (p. 11), where the original proof discovered (independently) by Bourbaki and Witt in 1950 is attributed to Serge Lang's Algebra. Again, the cartesian closedness of the category of compactly generated spaces is attributed to Steenrod, not to Ronnie Brown who got there first. (Goubault-Larrecq is by no means the first author to commit either of these solecisms, but that is no excuse.) But these criticisms should not be allowed to obscure the book's solid virtues. It is well written, and profusely (and helpfully!) illustrated. The notation is well-chosen, and the numerous exercises are well-integrated into the text, so that it would make a good self-study text. I do not imagine that many pure mathematicians will wish to adopt it as a course text for courses on point-set topology, but it would make a very useful alternative reference for students who wish to explore aspects of the subject not usually taught in such courses, provided they were told not to take its attributions on trust! © 2014 London Mathematical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Non-Hausdorff topology and domain theory: selected topics in point-set topology (New Mathematical Monographs 22) BY Jean Goubault-Larrecq

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Publisher
Oxford University Press
Copyright
© 2014 London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdu099
Publisher site
See Article on Publisher Site

Abstract

Many pure mathematicians (indeed, many topologists ) would be surprised to learn just how much point-set topology is used these days by theoretical computer scientists. The trend began in 1972 with the introduction by Dana Scott of the topology that bears his name, initially on a continuous lattice but ultimately on an arbitrary directed-complete poset, and the nearly 500 pages of this volume bear eloquent testimony to the extent of the development that has followed. However, one feature of the topologies employed in computer science is that many of them are non-Hausdorff, which means that traditional textbooks on topology, with their emphasis on Hausdorff spaces, are less than helpful to computer science students. It is to counter this bias that the present book has been written. (However, the author overstates his case when he says on p. 1, and again on p. 49 when he introduces the Hausdorff axiom, that ‘Most traditional topology textbooks assume the Hausdorff separation condition from the very start’; in a quick survey of the textbooks on my shelves, I could not find one that does this, and many introduce the axiom substantially later than Goubault-Larrecq does. Also, despite the title, there are Hausdorff spaces in the book: indeed, after a brief summary of set theory, the first substantial chapter is devoted to metric spaces.) Nevertheless, the emphasis throughout this book is on topics that receive only fleeting attention, if any, in ‘mainstream’ topology texts: compactly generated spaces (in the non-Hausdorff sense) and cartesian closedness, quasi-metric spaces and their various notions of completeness, sober spaces and frames, stable local compactness, powerdomains and so on. Although, as already indicated, the motivation for studying this material is applications in computer science, one does not need to know any computer science in order to read the book; some applications to domain theory appear towards the end of it, but they are treated from a purely mathematical viewpoint. The reference to cartesian closedness above indicates that categorical ideas play a larger rôle in the book than they do in many texts on topology. However, one thing that I personally found irritating was that, having collected all the material he needs from set theory and ordered-set theory in a preliminary chapter, Goubault-Larrecq introduces the categorical ideas in dribs and drabs, as and when he needs them; surely it would have been neater to put this material at the beginning, too? And I would not recommend this book to anyone wishing to learn about frame theory; what it says is too little and too late in the development to be helpful. The author has a disconcerting habit of naming almost every result of significance after its supposed author(s). I was in particular intrigued to discover the identity of the ‘Johnstone space’ (p. 143) and of ‘Johnstone's theorem’ (p. 427): both are in fact due to me, but I would not count them among my most profound achievements. And, in general, the assignment of credit for ideas is distinctly haphazard: references tend to be to textbooks rather than original papers (it is significant that Scott's path-breaking 1972 paper, mentioned earlier, is not in the bibliography), with the result that the textbook authors tend to get credit for other people's ideas. A good example occurs in the very first substantial result proved in the book, the Bourbaki–Witt fixed-point theorem (p. 11), where the original proof discovered (independently) by Bourbaki and Witt in 1950 is attributed to Serge Lang's Algebra. Again, the cartesian closedness of the category of compactly generated spaces is attributed to Steenrod, not to Ronnie Brown who got there first. (Goubault-Larrecq is by no means the first author to commit either of these solecisms, but that is no excuse.) But these criticisms should not be allowed to obscure the book's solid virtues. It is well written, and profusely (and helpfully!) illustrated. The notation is well-chosen, and the numerous exercises are well-integrated into the text, so that it would make a good self-study text. I do not imagine that many pure mathematicians will wish to adopt it as a course text for courses on point-set topology, but it would make a very useful alternative reference for students who wish to explore aspects of the subject not usually taught in such courses, provided they were told not to take its attributions on trust! © 2014 London Mathematical Society

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Dec 17, 2014

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