索书号：0189/M966(2) (MIT)

Topology

Contents

Preface

A note to the reader

Part I GENERAL TOPOLOGY

Chapter 1 set theory and logical

  1. fundamental concepts
  2. functions
  3. relations
  4. the integers and the real numbers
  5. Cartesian products
  6. finite sets
  7. countable and uncountable sets
  8. the principle of recursive definition
  9. infinite sets and the axiom of choice
  10. well-ordered sets
  11. the maximum principle

Chapter 2 topological spaces and continunous functions

  1. topological spaces
  2. basis for a topology
  3. the order topology
  4. the product topology on x*y
  5. the subspace topology
  6. closed sets and limit points
  7. continuous functions
  8. the product topology
  9. the metric topology
  10. the metric topology (continued)
  11. the quotient topology

Chapter 3 connectedness and compactness

  1. connected spaces
  2. connected subspaces of the real line
  3. components and local connectedness
  4. compact spaces
  5. compact subspaces of the real line
  6. limit point compactness
  7. local compactness

Chapter 4 countability and separation axioms

  1. the countability axioms
  2. the separation axioms
  3. normal spaces
  4. the urysohn lemma
  5. the urysohn metrization theorem
  6. the tietze extension theorem
  7. imbeddings of manifolds

Chapter 5 the tychonoff theorem

  1. the tychonoff theorem
  2. the stone-cech compactification

Chapter 6 metization theorems and paracompactness

  1. local finiteness
  2. the nagata-smirnov metrization theorem
  3. paracompactness
  4. the smirnov metrization theorem

Chapter 7 complete metric spaces and function spaces

  1. complete metric spaces
  2. a space-filling curve
  3. compactness in metric spaces
  4. pointwise and compact convergence
  5. ascoli’s theorem

Chapter 8 baire spaces and dimension theory

  1. baire spaces
  2. a nowhere differentiable function
  3. introduction to dimension theory

part II ALGEBRAIC TOPOLOGY

Chapter 9 the fundamental group

  1. homotopy of paths
  2. the fundamental group
  3. covering spaces
  4. the fundamental group of the circle
  5. retractions and fixed points
  6. the fundamental theorem of algebra
  7. the borsuk-ulam theorem
  8. deformation retracts and homotopy type
  9. the fundamental group of sn
  10. fundamental groups of some surfaces

Chapter10 separation theorems in the plane

  1. the Jordan separation theorem
  2. invariance of domain
  3. the Jordan curve theorem
  4. imbedding graphs in the plane
  5. the winding number of a simple closed curve
  6. the Cauchy integral formula

Chapter 11 the seifert-van kampen theorem

  1. direct sums of abelian groups
  2. free products of groups
  3. free groups
  4. the seifert-van kampen theorem
  5. the fundamental group of a wedge of circles
  6. adjoining a two-cell
  7. the fundamental groups of the torus and the dunce cap

Chapter 12 classification of surfaces

  1. fundamental groups of surfaces
  2. homology of surfaces
  3. cutting and pasting
  4. the classification theorem
  5. constructing compact surfaces

Chapter 13 classification of covering spaces

  1. equivalence of covering spaces
  2. the universal covering space
  3. covering transformations
  4. existence of covering spaces

Chapter 14 applications to group theory

  1. covering spaces of a graph
  2. the fundamental group o f a graph
  3. subgroups of free groups

Abstract

This book is intended as a text for a one- or two-semester introduction to topology, at the senior or first-year graduate level. It can be used for a number of different courses.

Part I, consisting of the first eight chapters,is devoted to the subject commonly called general topology. Part II constitutes an introduction to the subject of Algebraic Topology.