Book:Gerard Buskes/Topological Systems: From Distance to Neighborhood

Subject Matter

 * Topology
 * Metric Spaces

Contents

 * Preface


 * PART I. THE LINE AND THE PLANE
 * Chapter 1. What Topology Is About


 * Topological Equivalence
 * Continuity and Convergence
 * A Few Conventions
 * Extra: Topological Diversions
 * Exercises


 * Chapter 2. Axioms for $\R$


 * Extra: Axiom Systems
 * Exercises


 * Chapter 3. Convergent Sequences and Continuity


 * Subsequences
 * Uniform Continuity
 * The Plane
 * Extra: Bolzano (1781-1848)
 * Exercises


 * Chapter 4. Curves in the Plane


 * Curves
 * Homeomorphic Sets
 * Brouwer's Theorem
 * Extra: L.E.J. Brouwer (1881-1966)


 * PART II. METRIC SPACES
 * Chapter 5. Metrics


 * Extra: Camille Jordan (1838-1922)
 * Exercises


 * Chapter 6. Open and Closed Sets


 * Subsets of a Metric Space
 * Collections of Sets
 * Similar Metrics
 * Interior and Closure
 * The Empty Set
 * Extra: Cantor (1845-1918)


 * Chapter 7. Completeness


 * Extra: Meager Sets and the Mazur Game
 * Exercises


 * Chapter 8. Uniform Convergence


 * Extra: Spaces of Continuous Functions
 * Exercises


 * Chapter 9. Sequential Compactness


 * Extra: The p-adic Numbers
 * Exercises


 * Chapter 10. Convergent Nets


 * Inadequacy of Sequences
 * Convergent Nets
 * Extra: Knots
 * Exercises


 * Chapter 11. Transition to Topology


 * Generalized Convergence
 * Topologies
 * Extra: The Emergence of the Professional Mathematician
 * Exercises


 * PART III. TOPOLOGICAL SPACES
 * Chapter 12. Topological Spaces


 * Extra: Map Coloring
 * Exercises


 * Chapter 13. Compactness and the Hausdorff Property


 * Compact Spaces
 * Hausdorff Spaces
 * Extra: Hausdorff and the Measure Problem
 * Exercises


 * Chapter 14. Products and Quotients


 * Product Spaces
 * Quotient Spaces
 * Extra: Surfaces
 * Exercises


 * Chapter 15. The Hahn-Tietze-Tong-Urysohn Theorems


 * Urysohn's Lemma
 * Interpolation and Extension
 * Extra: Nonstandard Mathematics
 * Exercises


 * Chapter 16. Connectedness


 * Connected Spaces
 * The Jordan Theorem
 * Extra: Continuous Deformation of Curves
 * Exercises


 * Chapter 17. Tychonoff's Theorem


 * Extra: The Axiom of Choice
 * Exercises


 * PART IV. POSTSCRIPT
 * Chapter 18. A Smorgasbord for Further Study


 * Countability Conditions
 * Separation Conditions
 * Compactness Conditions
 * Compactifications
 * Connectivity Conditions
 * Extra: Dates from the History of General Topology
 * Exercises


 * Chapter 19. Countable Sets


 * Extra: The Continuum Hypothesis