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Counterexamples in Topology

  • Lynn Arthur Steen
  • J. Arthur SeebachJr.

Table of contents

  1. Front Matter
    Pages i-xi
  2. Basic Definitions

    1. Front Matter
      Pages 1-1
    2. Lynn Arthur Steen, J. Arthur Seebach Jr.
      Pages 3-10
    3. Lynn Arthur Steen, J. Arthur Seebach Jr.
      Pages 11-17
    4. Lynn Arthur Steen, J. Arthur Seebach Jr.
      Pages 18-27
    5. Lynn Arthur Steen, J. Arthur Seebach Jr.
      Pages 28-33
    6. Lynn Arthur Steen, J. Arthur Seebach Jr.
      Pages 34-38
  3. Counterexamples

    1. Front Matter
      Pages 39-39
    2. Lynn Arthur Steen, J. Arthur Seebach Jr.
      Pages 39-158
  4. Metrization Theory

    1. Front Matter
      Pages 159-159
    2. Lynn Arthur Steen, J. Arthur Seebach Jr.
      Pages 161-181
  5. Back Matter
    Pages 183-244

About this book

Introduction

The creative process of mathematics, both historically and individually, may be described as a counterpoint between theorems and examples. Al­ though it would be hazardous to claim that the creation of significant examples is less demanding than the development of theory, we have dis­ covered that focusing on examples is a particularly expeditious means of involving undergraduate mathematics students in actual research. Not only are examples more concrete than theorems-and thus more accessible-but they cut across individual theories and make it both appropriate and neces­ sary for the student to explore the entire literature in journals as well as texts. Indeed, much of the content of this book was first outlined by under­ graduate research teams working with the authors at Saint Olaf College during the summers of 1967 and 1968. In compiling and editing material for this book, both the authors and their undergraduate assistants realized a substantial increment in topologi­ cal insight as a direct result of chasing through details of each example. We hope our readers will have a similar experience. Each of the 143 examples in this book provides innumerable concrete illustrations of definitions, theo­ rems, and general methods of proof. There is no better way, for instance, to learn what the definition of metacompactness really means than to try to prove that Niemytzki's tangent disc topology is not metacompact. The search for counterexamples is as lively and creative an activity as can be found in mathematics research.

Keywords

Compactification Connected space Separation axiom Topologie metrizable topology

Authors and affiliations

  • Lynn Arthur Steen
    • 1
  • J. Arthur SeebachJr.
    • 1
  1. 1.Saint Olaf CollegeNorthfieldUSA

Bibliographic information