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 Appendixes: 1 Examples of metric spaces; 2 Set theory and algebra; 3 The transition to topological spaces Selected bibliography Index.
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
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