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Set Theory and Metric Spaces
Irving Kaplansky

AMS Chelsea Publishing
1972; 140 pp; hardcover
Volume: 298
ISBN-10: 0-8218-2694-8
ISBN-13: 978-0-8218-2694-2
List Price: US$32
Member Price: US$28.80
Order Code: CHEL/298.H
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This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index.

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Undergraduates and graduate students interested in set theory and metric spaces.


"This is a book that could profitably be read by many graduate students or by seniors in strong major programs ... has a number of good features. There are many informal comments scattered between the formal development of theorems and these are done in a light and pleasant style. ... There is a complete proof of the equivalence of the axiom of choice, Zorn's Lemma, and well-ordering, as well as a discussion of the use of these concepts. There is also an interesting discussion of the continuum problem ... The presentation of metric spaces before topological spaces ... should be welcomed by most students, since metric spaces are much closer to the ideas of Euclidean spaces with which they are already familiar."

-- Canadian Mathematical Bulletin

"Kaplansky has a well-deserved reputation for his expository talents. The selection of topics is excellent."

-- Lance Small, UC San Diego

Table of Contents

Basic Set Theory
  • 1.1 Inclusion; 1.2 Operations on sets; 1.3 Partially ordered sets and lattices; 1.4 Functions; 1.5 Relations; Cartesian products
Cardinal Numbers
  • 2.1 Countable sets; 2.2 Cardinal numbers; 2.3 Comparison of cardinal numbers; Zorn's lemma; 2.4 Cardinal addition; 2.5 Cardinal multiplication; 2.6 Cardinal exponentiation
Well-Ordering; The Axiom of Choice
  • 3.1 Well-ordered sets; 3.2 Ordinal numbers; 3.3 The axiom of choice; 3.4 The continuum problem
Basic Properties of Metric Spaces
  • 4.1 Definitions and examples; 4.2 Open sets; 4.3 Convergence; Closed sets; 4.4 Continuity
Completeness, Separability, and Compactness
  • 5.1 Completeness; 5.2 Separability; 5.3 Compactness
Additional Topics
  • 6.1 Product spaces; 6.2 A fixed-point theorem; 6.3 Category
  • 1 Examples of metric spaces; 2 Set theory and algebra; 3 The transition to topological spaces
  • Selected bibliography
  • Index
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