Introduction to Topology

Introduction to Topology

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One of the most important milestones in mathematics in the twentieth century was the development of topology as an independent field of study and the subsequent systematic application of topological ideas to other fields of mathematics. While there are many other works on introductory topology, this volume employs a methodology somewhat different from other texts. Metric space and point-set topology material is treated in the first two chapters; algebraic topological material in the remaining two. The authors lead readers through a number of nontrivial applications of metric space topology to analysis, clearly establishing the relevance of topology to analysis. Second, the treatment of topics from elementary algebraic topology concentrates on results with concrete geometric meaning and presents relatively little algebraic formalism; at the same time, this treatment provides proof of some highly nontrivial results. By presenting homotopy theory without considering homology theory, important applications become immediately evident without the necessity of a large formal program. Prerequisites are familiarity with real numbers and some basic set theory. Carefully chosen exercises are integrated into the text (the authors have provided solutions to selected exercises for the Dover edition), while a list of notations and bibliographical references appear at the end of the more

Product details

  • Paperback | 256 pages
  • 152.4 x 228.6 x 15.24mm | 158.76g
  • Dover Publications Inc.
  • New York, United States
  • English
  • Revised
  • 2nd Revised edition
  • 0486406806
  • 9780486406800
  • 320,162

Table of contents

ONE METRIC SPACES   1 Open and closed sets   2 Completeness   3 The real line   4 Products of metric spaces   5 Compactness   6 Continuous functions   7 Normed linear spaces   8 The contraction principle   9 The Frechet derivative TWO TOPOLOGICAL SPACES   1 Topological spaces   2 Subspaces   3 Continuous functions   4 Base for a topology   5 Separation axioms   6 Compactness   7 Locally compact spaces   8 Connectedness   9 Path connectedness   10 Finite product spaces   11 Set theory and Zorn's lemma   12 Infinite product spaces   13 Quotient spaces THREE HOMOTOPY THEORY   1 Groups   2 Homotopic paths   3 The fundamental group   4 Induced homomorphisms   5 Covering spaces   6 Some applications of the index   7 Homotopic maps   8 Maps into the punctured plane   9 Vector fields   10 The Jordan Curve Theorem FOUR HIGHER DIMENSIONAL HOMOTOPY   1 Higher homotopy groups   2 Noncontractibility of Sn   3 Simplexes and barycentric subdivision   4 Approximation by piecewise linear maps   5 Degrees of maps   BIBLIOGRAPHY   LIST OF NOTATIONS   SOLUTIONS TO SELECTED EXERCISES   INDEXshow more

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21 ratings
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