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Discover a unique and modern treatment of topology employing a cross-disciplinary approach
Implemented recently to understand diverse topics, such as cell biology, superconductors, and robot motion, topology has been transformed from a theoretical field that highlights mathematical theory to a subject that plays a growing role in nearly all fields of scientific investigation. Moving from the concrete to the abstract, Topology and Its Applications displays both the beauty and utility of topology, first presenting the essentials of topology followed by its emerging role within the new frontiers in research.
Filling a gap between the teaching of topology and its modern uses in real-world phenomena, Topology and Its Applications is organized around the mathematical theory of topology, a framework of rigorous theorems, and clear, elegant proofs.
This book is the first of its kind to present applications in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, and population modeling, as well as other areas of science and engineering. Many of these applications are presented in optional sections, allowing an instructor to customize the presentation.
The author presents a diversity of topological areas, including point-set topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include:
- Open sets
- Compactness
- Homotopy
- Surface classification
- Index theory on surfaces
- Manifolds and complexes
- Topological groups
- The fundamental group and homology
Special "core intuition" segments throughout the book briefly explain the basic intuition essential to understanding several topics. A generous number of figures and examples, many of which come from applications such as liquid crystals, space probe data, and computer graphics, are all available from the publisher's Web site.
- Sales Rank: #1713266 in Books
- Published on: 2006-11-03
- Original language: English
- Number of items: 1
- Dimensions: 9.50" h x .90" w x 6.30" l, 1.46 pounds
- Binding: Hardcover
- 384 pages
Review
"…helpful to a beginning student, especially one who is interested in the connections between topology and the world of applications." (Mathematical Reviews, 2007k)
"…a welcome addition to what is now a long list of good undergraduate topology books." (CHOICE, August 2007)
"..a celebration of topology and its many applications. I enjoyed reading it and believe that it would be an interesting textbook from which to learn." (MAA Reviews, January 12, 2007)
From the Back Cover
Discover a unique and modern treatment of topology employing a cross-disciplinary approach
Implemented recently to understand diverse topics, such as cell biology, superconductors, and robot motion, topology has been transformed from a theoretical field that highlights mathematical theory to a subject that plays a growing role in nearly all fields of scientific investigation. Moving from the concrete to the abstract, Topology and Its Applications displays both the beauty and utility of topology, first presenting the essentials of topology followed by its emerging role within the new frontiers in research.
Filling a gap between the teaching of topology and its modern uses in real-world phenomena, Topology and Its Applications is organized around the mathematical theory of topology, a framework of rigorous theorems, and clear, elegant proofs.
This book is the first of its kind to present applications in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, and population modeling, as well as other areas of science and engineering. Many of these applications are presented in optional sections, allowing an instructor to customize the presentation.
The author presents a diversity of topological areas, including point-set topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include:
- Open sets
- Compactness
- Homotopy
- Surface classification
- Index theory on surfaces
- Manifolds and complexes
- Topological groups
- The fundamental group and homology
Special "core intuition" segments throughout the book briefly explain the basic intuition essential to understanding several topics. A generous number of figures and examples, many of which come from applications such as liquid crystals, space probe data, and computer graphics, are all available from the publisher's Web site.
About the Author
WILLIAM F. BASENER, PhD, is an Assistant Professor in the Department of Mathematics and Statistics at Rochester Institute of Technology in Rochester, New York. He received his PhD in mathematics in 2001 from Boston University. His research interests include dynamical systems, differential equations, applied topology, economics, and the topology of manifolds. Dr. Basener is the recipient of numerous teaching awards.
Most helpful customer reviews
19 of 21 people found the following review helpful.
Needs corrections?
By Christopher Grant
A student came by my office yesterday with a question about an exercise from this book that asked the reader to show that the product topology and the standard metric topology on R^2 are equivalent. The student thought he had a counterexample (a disc containing part of its boundary), and he was right as far Basener's faulty definition of the product topology was concerned. Basener defines a set to be open in the product topology if and only if its image under each projection is open.
I scoured the Internet for mention of this error and came across the Zentralblatt review (which is omitted from Amazon's list of editorial reviews for this book). That review lists this error and others and states in summary: "The book is absolutely terrible."
That's obviously a strong assertion, and, not having read the book, I'm in no position to confirm or deny it. Still, I felt it was important to post this note to warn potential purchasers/readers of problems with this text, and to encourage the author and publisher to fix things. Neither the publisher website nor the (apparently broken) author website nor a Google search yielded a list of errata.
[8/19/2008 Update: The author's website has been fixed, and it has a list of errata. Furthermore, the author assures me that the error in the definition of the product topology has been fixed in printings after the first.]
0 of 0 people found the following review helpful.
Good explanations and examples, faulty proofs
By C. Pettis
On the upside, the author strives and does a good job to convey intuition and visual examples, which I think is important and not enough people do this, so kudos to the author for this.
On the downside, there are many errors in the exposition and proofs. They range from innocuous (pg. 20, Lemma 6 says 'union' when the equation clearly and correctly shows 'intersection') to intermediate (pg 28, Theorem 9, he claims 'f(f^-1(B)) = B' when really the lhs is a subset of the rhs), to ones that threw me off (pg. 24, 'U x V = p_1(U) \intersect p_2(V)' should be 'U x V = p_1^-1(U) \intersect p_2^-1(V)'; and pg 65, theorem 31, has incorrect / inconsistent variable references, and states without proof that open balls are path connected. That last one isn't an error, but I had to look up, and this author, who is careful with the concepts part, seems like it would be his style to include it).
All in all, I could recommend a future version if the errors are corrected. It gave me insights and had a conversational tone that made it easy to learn from, but I had to stop because the errors threw me off too much to have confidence that I'll catch more subtle problems in more unfamiliar territory. I hope the author does an updated version with corrections.
0 of 0 people found the following review helpful.
Visualized Topology
By Vic Dannon
The book is unique in its attempt to visualize Topology.
The Schaum's outline on Topology may be all needed to pass an exam,
But this book has a lot more figures and applications than many topology books.
Colored figures would improve it.
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