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Introduction to Topology and Modern Analysis Paperback – January 1, 1963


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Paperback, January 1, 1963
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Product Details

  • Paperback: 372 pages
  • Publisher: McGraw-Hill; International Student Edition edition (1963)
  • ASIN: B000OG1OB4
  • Product Dimensions: 5.8 x 0.7 x 8.2 inches
  • Average Customer Review: 4.9 out of 5 stars  See all reviews (12 customer reviews)
  • Amazon Best Sellers Rank: #10,341,865 in Books (See Top 100 in Books)

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All in all, a highly recommended intro to the subject.
"jsurti"
Groups, rings, and fields are given a minimal treatment by the author, discussing only the basic rudiments that are needed to get through the rest of the book.
Dr. Lee D. Carlson
It is always the book that first comes to mind when someone asks for a reference on any of the subjects it covers.
Kevin R. Vixie

Most Helpful Customer Reviews

57 of 59 people found the following review helpful By Dr. Lee D. Carlson HALL OF FAMEVINE VOICE on July 5, 2002
Format: Hardcover
In the author's words in the preface, the dominant theme of this book is continuity and linearity, and its goal is to illuminate the meanings of these words and their relations to each other. The book, he says, belongs to the type of pure mathematics that is concerned with form and structure, and such a body of mathematics must be judged by its high aesthetic quality, and should exalt the mind of the reader.
The author's attitude can only be characterized as magnificent, and, if one is to judge his utterances in the preface by what is found after it, one will indeed find perfect evidence of his delight in mathematics and his high competence in elucidating very abstract concepts in topology and real analysis. Indeed, this has to be the best book ever written for mathematics at this level. It is a book that should be read by everyone that desires deep insights into modern real and functional analysis.
After a brief and informal overview of set theory, the author moves on to the theory of metric spaces in chapter 2. His emphasis is on the idea that metric spaces are easy to find, since every non-empty set has the discrete metric, and that metric spaces are good motivation for the more general idea of a topological space. The Cantor set, ubiquitous in measure theory, dynamical systems, and fractal geometry, is constructed as the most general closed set on the real line, i.e. one obtained by removing from the real line a countable disjoint class of open intervals. Continuity of mappings between metric spaces is defined, and also the concept of uniform continuity, the latter of which is motivated very nicely by the author.
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32 of 32 people found the following review helpful By Kevin R. Vixie on May 28, 2000
Format: Hardcover
I became aquainted with this book many years ago and I still read it ... and send students off to read it. The book is written by an incredible expositor who was and still may be at Colorado College in Colorado. It is always the book that first comes to mind when someone asks for a reference on any of the subjects it covers. These include point set topology, analysis (Not including integration or measure theory), and operator theory. It is introductory. This merely makes you wish the author would have written several advanced sequels to this amazing book.
This book has my highest recommendation. Every mathematics student should own a copy ...
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15 of 15 people found the following review helpful By A. Ali on February 19, 2002
Format: Hardcover
This is a fine book, but not quite in the 5-star league. Let me elaborate. The book is divided into three parts: general topology, the theory of Banach and Hilbert spaces, and Banach algebras. The first two parts lead, by way of synthesis, to the last part, where some interesting but elementary results are proved about Banach algebras in general and C*-algebras in particular. I might mention, for example, the Spectral theorem for compact self-adjoint operators, the Stone representation theorem, and the Gelfand-Naimark theorem.
I can attest from personal experience that the book is well-written; indeed I worked through it chapter by chapter. But today there do exist a plethora of other treatments that can at least rival this text in lucidity, organisation and coverage. For example, for general topology, there is an excellent text by Willard titled 'General Topology',as well as Hocking and Young's old 'Topology'. Both of these go much further in the realm of point-set topology than Simmons. Similarly there are any number of well-written texts on functional analysis that cover the subject of Banach spaces, Hilbert spaces and self-adjoint operators very clearly. Indeed in some respects I feel the Simmons book was inadequate by itself and needed to be supplemented by a text on linear algebra; self-adjoint operators -- and by implication, the Spectral theorem -- need to be seen and manipulated in the finite-dimensional version before one examines their infinite-dimensional generalisation. The Simmons book is a bit weak here; one needs to be playing with matrices.
These are, however, minor quibbles. The book can be recommended to a junior- or senior-level undergraduate.
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18 of 19 people found the following review helpful By "jsurti" on November 6, 2003
Format: Hardcover
The first part of this book that deals with topology is a pedagogical masterpiece. After motivating the key concepts of compactness and continuity in the relatively concrete setting of metric spaces, the book goes on to abstract topological spaces, a beautiful section on compactness including the tychonoff theorem, and an extremely lucid development of the separation axioms and the proof of the urysohn imbedding theorem and the stone-cech compactification. I personally find the chapter on connectedness to be the weak link in this part of the book. Wherever possible, Simmons provides an exhaustive list of examples (especially when introducing the various types of spaces) that aids comprehension. Moreover, some of the central concepts (product topology) and deeper results such as the Stone-Cech compactification are easier to appreciate because the author has a section on topological properties of the relevant function spaces couple of chapters ahead and several exercises along the way. All in all, a highly recommended intro to the subject.
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