Set Theory

Presents various theorems of the theory of sets along with complete proofs. This book discusses strengthening of theorems, the simplification of proofs, and the removal of unnecessary hypotheses.

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.