[Book Cover]

Elementary Topology, 1/e

Dennis Roseman, University of Iowa

Published June, 1999 by Prentice Hall Engineering/Science/Mathematics

Copyright 1999, 400 pp.
Cloth
ISBN 0-13-863879-9


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Summary

For a one-semester, advanced undergraduate level course in Introduction to Topology. Designed both for students who will take only one course in topology as well as for those who are preparing for more advanced work, this text offers a thorough introduction to the important topics of topology, a variety of interesting, concrete examples, and ample opportunity and guidance for building reasoning skills and writing proofs. It integrates students' background in calculus, analytic geometry and linear algebra throughout the presentation.

Features


Presents a selection of interesting advanced topics — e.g., manifolds, complexes, knots— in a manner undergraduates can handle.

  • Some basic mathematical results derived from the basic core material are included.
Provides an extra, “guided index” — an extensive list of topics collected as Examples, Propositions and Remarks — designed to help students solve problems and investigate examples.
Features a less-abstract, more example-driven approach that uses the setting of Euclidean space — mostly the line, the plane and 3-dimensional space.
  • Focuses on questions, e.g.: What are the different subsets of the plane (or space) and what should “different” mean?
  • Considers subsets of Rn. Demonstrates the need for a tool before presenting it.
  • Uses geometry, analytic geometry, linear algebra, calculus, and complex numbers to describe subsets and continuous functions — and to show that topology is not isolated from things about which students already know.
Focuses on Rn — to get more quickly and more in-depth into more interesting problems (rather than getting into long discussions of separation properties, bases and sub-bases, countability).
  • Presents topics of general topology — not relevant to Rn — in an appendix.
Emphasizes examples.
  • Develops concepts to answer questions about some subsets of Euclidean space.
  • Several proofs are given to verify a single relationship.
Explores basic mathematical questions — e.g., What are the subsets of R2? What are the possibilities of embedding one object into another?
Offers some novel alternate approaches to standard problems, e.g.:
  • The first proof offered that R1 is not homeomorphic to R2 is that there are non-equivalent embeddings of a three point set into R1, but all embeddings of a three point set into R2 are equivalent.
  • The first proof that the interval, I, is not homeomorphic to the circle, S1, is that I has the fixed point property, S1 does not.


Table of Contents
    Preface
    I. BASIC TOPICS.
    1. Open and Closed Subsets.
    2. Building Open and Closed Subsets.
    3. Continuity.
    4. Homeomorphism.
    5. Cantor Sets and Allied Topics.
    6. Embeddings.
    7. Connectivity .
    8. Path Connectedness.
    9. Closure and Limit Points.
    10. Compactness.
    11. Local Connectivity.
    II. ADVANCED TOPICS
    12. Space Filling Curves.
    13. Manifolds.
    14. Knots and Kottings.
    15. Simple Connectivity.
    16. Deformation Type.
    17. Complexes.
    18. Higher Dimensions.
    19. The Poincaré Conjecture.
    III. APPENDICES.
    Appendix A. Sets and Logic.
    Appendix B. Numbers.
    Appendix C. Cardinality of Sets.
    Appendix D. Summary from Calculus.
    Appendix E. Strategy in Proof.
    Bibliography.
    Index of Examples, Remarks, and Propositions.
    Subject Index.


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