文档介绍：INTRODUCTORY
REAL ANALYSIS
A. N. KOLMOGOROV
S. V. FOMIN
Revised English Edition
Translated and Edited by
Richard A. Silverman
DOVER PUBLICATIONS, INC.
NEW YORK
CONTENTS
1 SET THEORY, Page 1.
1. Sets and Functions, 1.
. Basic definitions, 1.
. Operations on sets, 2.
. Functions and mappings. Images and preimages,
4.
. position of a set into classes. Equivalence
relations, 6.
2. Equivalence of Sets. The Power of a Set, 9
. Finite and infinite sets, 9.
. Countable sets, 10.
. Equivalence of sets, 13.
. Uncountability of the real numbers, 14.
. The power of a set, 16.
. The Cantor-Bernstein theorem, 17.
3. Ordered Sets and Ordinal Numbers, 20.
. Partially ordered sets, 20.
. Order-preserving mappings. Isomorphisms, 21.
. Ordered sets. Order types, 21.
. Ordered sums and products of ordered sets, 22.
. Well-ordered sets. Ordinal numbers, 23.
. Comparison of ordinal numbers, 25.
. The well-ordering theorem, the axiom of choice
and equivalent assertions, 27.
. Transfinite induction, 28.
. Historical remarks, 29.
4. Systems of Sets, 31.
. Rings of sets, 31.
. Semirings of sets, 32.
. The ring generated by a semiring, 34.
. Borel algebras, 35.
2 METRIC SPACES, Page 37.
5. Basic Concepts, 37.
. Definitions and examples, 37.
. Continuous mappings and homeomorphisms.
Isometric spaces, 44.
vii
viii CONTENTS
6. Convergence. Open and Closed Sets, 45. . Continuous and semicontinuous functions on
. Closure of a set. Limit points, 45. compact spaces, 109.
. Convergence and limits, 47. . Continuous curves in metric spaces, 112.
, bense subsets. Separable spaces, 48.
. Closed sets, 49.
. Open sets, 50. LINEAR SPACES, Page 118.
. Open and closed sets on the real line, 51.
7. Complete Metric Spaces, 56. 13. Basic Concepts, 118.
. Defin