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A Guide to Topology

1 rating: 5.0
A book by Steven G. Krantz

The intention of the guides is to support graduate education in mathematics by providing thumbnail sketches of subject areas that a student might use a a starting point or as a review....The author's depth in anallysis is apparent in his choices … see full wiki

Tags: Books, Mathematics, Topology
Author: Steven G. Krantz
Publisher: Mathematical Association of America
1 review about A Guide to Topology

If your need is an overview/review of point-set topology, then this is the book for you

  • Dec 4, 2009
A study of topology is an integral part of the education of most graduate students and knowledge of topology is an essential skill for theoretical physics and the essential topology of a network is discussed in introductory computer classes. This book is designed to be a topology handbook rather than a text, it begins with the basics and given the assumption of significant mathematical maturity, moves very quickly through the essentials of point-set topology.
There are no exercises and the section headings are:

Main heading: Fundamentals
*) What is topology?
*) First definitions
*) Mappings
*) The separation axioms
*) Compactness
*) Homeomorphisms
*) Connectedness
*) Path-connectedness
*) Continua
*) Totally disconnected sets
*) The Cantor set
*) Metric spaces
*) Metrizability
*) Baire's theorem
*) Lebesgue's lemma and Lebesgue numbers

Main heading: Advanced properties of topological spaces
*) Basis and subbasis
*) Product spaces
*) Relative topology
*) First countable and second countable
*) Compactifications
*) Quotient topologies
*) Uniformities
*) Morse theory
*) Proper mappings
*) Paracompactness

Main heading: Moore-Smith convergence and nets
*) Introductory remarks
*) Nets

Main heading: Function spaces
*) Preliminary ideas
*) The topology of pointwise convergence
*) The compact-open topology
*) Uniform convergence
*) Equicontinuity and the Ascoli-Arzela theorem
*) The Weirstrass approximation theorem

Given that the actual text is 85 pages and the introductory section is 46 pages in length, the treatment of each of the advanced topics is rather short, in many cases less than two pages. Therefore, brevity is the operative word, while Krantz is an excellent expository writer, there simply is no space for any depth or a large number of detailed examples. In other words, the coverage is very compact. This is an excellent text if your need is for an overview or review, of little value if you need an in-depth introduction to the topic.

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