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Full Description
In the early twentieth century, Hausdorff developed an axiomatic approach to topology, which continues to be the foundation of modern topology. The present book, the English translation of the third edition of Hausdorff's Mengenlehre, is a thorough introduction to his theory of point-set topology.
The treatment begins with topics in the foundations of mathematics, including the basics of abstract set theory, sums and products of sets, cardinal and ordinal numbers, and Hausdorff's well-ordering theorem. The exposition then specializes to point sets, where major topics such as Borel systems, first and second category, and connectedness are considered in detail. Next, mappings between spaces are introduced. This leads naturally to a discussion of topological spaces and continuous mappings between them. Finally, the theory is applied to the study of real functions and their properties.
The book does not presuppose any mathematical knowledge beyond calculus, but it does require a certain maturity in abstract reasoning; qualified college seniors and first-year graduate students should have no difficulty in making the material their own.
Contents
Sets and the Combining of Sets: 1.1 Sets
1.2 Functions
1.3 Sum and intersection
1.4 Product and power
Cardinal Numbers: 2.5 Comparison of sets
2.6 Sum, product, and power
2.7 The scale of cardinal numbers
2.8 The elementary cardinal numbers
Order Types: 3.9 Order
3.10 Sum and product
3.11 The types $\aleph_0$ and $\aleph$
Ordinal Numbers: 4.12 The well-ordering theorem
4.13 The comparability of ordinal numbers
4.14 The combining of ordinal numbers
4.15 The alefs
4.16 The general concept of product
Systems of Sets: 5.17 Rings and fields
5.18 Borel systems
5.19 Suslin sets
Point Sets: 6.20 Distance
6.21 Convergence
6.22 Interior points and border points
6.23 The $\alpha, \beta$, and $\gamma$ points
6.24 Relative and absolute concepts
6.25 Separable spaces
6.26 Complete spaces
6.27 Sets of the first and second categories
6.28 Spaces of sets
6.29 Connectedness
Point Sets and Ordinal Numbers: 7.30 Hulls and kernels
7.31 Further applications of ordinal numbers
7.32 Borel and Suslin sets
7.33 Existence proofs
7.34 Criteria for Borel sets
Mappings of Two Spaces: 8.35 Continuous mappings
8.36 Interval-images
8.37 Images of Suslin sets
8.38 Homeomorphism
8.39 Simple curves
8.40 Topological spaces
Real Functions: 9.41 Functions and inverse image sets
9.42 Functions of the first class
9.43 Baire functions
9.44 Sets of convergence
Supplement: 10.45 The Baire condition
10.46 Half-schlicht mappings
Appendixes
Bibliography
Further references
Index
