Munkres Topology Chapter 3 Solutions

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Contents

Chapter 1. Set Theory and Logic

Chapter 2. Topological Spaces and Continuous 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 Point
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-Čech Compactification

Chapter 6. Metrization 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

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

Chapter 10. 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

Below are links to answers and solutions for exercises in the Munkres (2000) Topology, Second Edition.

Enjoy!