This work is a translation into English of the Third Edition of the classic German language work Mengenlehre by Felix Hausdorff published in 1937.

From the Preface (1937): "The present book has as its purpose an exposition of the most important theorems of the theory of sets, along with complete proofs, so that the reader should not find it necessary to go outside this book for supplementary details while, on the other hand, the book should enable him to undertake a more detailed study of the voluminous literature on the subject. The book does not presuppose any mathematical knowledge beyond the differential and integral 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 ... The mathematician will ... find in this book some things that will be new to him, at least as regards formal presentation and, in particular, as regards the strengthening of theorems, the simplification of proofs, and the removal of unnecessary hypotheses."

Readership

Graduate students and research mathematicians.

Reviews

"An indispensible book for all those interested in the theory of sets and the allied branches of real variable theory."

-- Bulletin of the AMS

Table of 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