This advanced textbook on topology has three unusual features. First, the introduction is from the locale viewpoint, motivated by the logic of finite observations: this provides a more direct approach than the traditional one based on abstracting properties of open sets in the real line. Second, the author freely exploits the methods of locale theory. Third, there is substantial discussion of some computer science applications. As computer scientists become more aware of the mathematical foundations of their discipline, it is appropriate that such topics are presented in a form of direct relevance and applicability. This book goes some way towards bridging the gap for computer scientists.
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Book Description
This is an advanced textbook on topology for computer scientists. It is based on a course given by the author to postgraduate students of computer science at Imperial College.
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17 of 18 people found the following review helpful:
5.0 out of 5 stars
Introducing locales, the algebraic equivalent of topologies.,
By A Customer
This review is from: Topology via Logic (Cambridge Tracts in Theoretical Computer Science) (Paperback)
This is a very accessible account relating the theory of locales to classical point set topology, with remarks along the way about the relation between the two theories in terms of category theory.Locales are lattices, which also have all joins of arbitrary subsets, just as topologies also have finite intersections of open sets and arbitrary unions; finite meets also distribute over arbitrary joins. In locale theory, you can do topology without using points (the art of "pointless thinking", as Johnstone likes to call it), although you can also introduce points in locales by definition. Locales, in other words, are complete lattices, and their elements can be identified with open sets. In the other direction, "points" in locales (which can be identified with principal prime ideals) correspond to the points in a corresponding topological space. The book is a nice introduction to Johnstone's now classic *Stone Spaces*. But the "logic&qu! ot; as in *Topology via Logic*-- refers to "logic" in the sense of computer science. Readers at home with symbolic logic may therefore find the examples, all based on "data streams" of bits, a bit strange, and may want to translate them into more familiar terms. But the reward is worth the effort. The informal style and free use of pictures to illustrate ideas is also a refreshing, and welcome change from the rigid and unnecessarily formal style too often encountered in books on this and similar topics.
5.0 out of 5 stars
Topology Via Logic-A nice bridge between Theory and reality,
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This review is from: Topology via Logic (Cambridge Tracts in Theoretical Computer Science) (Paperback)
The book Topology via Logic (Cambridge Tracts in Theoretical Computer Science) by Steven Vickers is the only book that I found a nice bridge between the abstract Topological theory and the general life reasoning. This not only a wonderful book for a Computer Scientists it is also a great book for Set Theoretic Topologists and Logicians who are always dealing with modelling natural language/incomplete knowledge etc. I found it is the best book to helping my research in incomplete information system with the the abstract Topological Theories.
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First Sentence:
The origins of topology are very different from the context we shall be working in, and it is probably as well to compare some different ideas of what it is. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
algebraic locale, spectral locale, compact coprime, reflective embeddings, spectral iff, spectral algebraic, coprime decomposition, finite locales, modulo frames, frame homomorphism, topological systems, algebraic dcpo, saturated compact, locale theory, finite meets, power locale, empty meet, compact points, spatial locales, categorically speaking, empty join, finite joins, upper closure, pointwise topology, affirmative assertions Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Hofmann-Mislove Theorem, Loc Spat, Proof First, Proof Suppose New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The origins of topology are very different from the context we shall be working in, and it is probably as well to compare some different ideas of what it is. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
algebraic locale, spectral locale, compact coprime, reflective embeddings, spectral iff, spectral algebraic, coprime decomposition, finite locales, modulo frames, frame homomorphism, topological systems, algebraic dcpo, saturated compact, locale theory, finite meets, power locale, empty meet, compact points, spatial locales, categorically speaking, empty join, finite joins, upper closure, pointwise topology, affirmative assertions Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Hofmann-Mislove Theorem, Loc Spat, Proof First, Proof Suppose New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 40
books:
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10
books
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- Stone Spaces (Cambridge Studies in Advanced Mathematics) by Peter T. Johnstone in Front Matter, page 4, and Back Matter
- General Topology (Graduate Texts in Mathematics) by John L. Kelley on page 4, and Back Matter
- Continuous Lattices by B. Banaschewski in Back Matter (1), and Back Matter (2)
- Mathematical Puzzles of Sam Loyd by Sam Loyd on page 4, and Back Matter
- Belief Revision (Cambridge Tracts in Theoretical Computer Science) by Peter Gärdenfors in Front Matter
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- Topological and Algebraic Structures in Fuzzy Sets: A Handbook of Recent Developments in the Mathematics of Fuzzy Sets (Trends in Logic) by S.E. Rodabaugh on page 56, page 187, and page 426
- Recent Trends in Data Type Specification: 10th Workshop on Specification of Abstract Data Types Joint with the 5th COMPASS Workshop, S. Margherita, ... Papers (Lecture Notes in Computer Science) by Egidio Astesiano on page 436
- Deductive Program Design (NATO ASI Series / Computer and Systems Sciences) on page 271
- Current Research in Operational Quantum Logic: Algebras, Categories, Languages (Fundamental Theories of Physics) by Bob Coecke on page 288
- Proceedings of the 7th & 8th Asian Logic Conferences: Hsi-Tou, Taiwan, 6-10 June 1999: Chongquing, China 29 August-2 September 2 002 by Rod Downey on page 397
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