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

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Gerard Buskes and Arnoud van Rooij: Topological Systems: From Distance to Neighborhood

Published $1997$, Springer-Verlag New York, Inc.

ISBN 0-387-94994-1.


Subject Matter


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