Problems and Theorems in Classical Set Theory

Operations on sets.- Countability.- Equivalence.- Continuum.- Sets of reals and real functions.- Ordered sets.- Order types.- Ordinals.- Ordinal arithmetic.- Cardinals.- Partially ordered sets.- Transfinite enumeration.- Euclidean spaces.- Zorn"s lemma.- Hamel bases.- The continuum hypothesis.- Ultrafilters on w.- Families of sets.- The Banach-Tarski paradox.- Stationary sets in w1.- Stationary sets in larger cardinals.- Canonical functions.- Infinite graphs.- Partition relations.- Triangle systems.- Set mappings.- Trees.- The measure problem.- Stationary sets.- The axiom of choice.- Well founded sets and the axiom of foundation.- Solutions: Operations on sets.- Countability.- Equivalence.- Continuum.- Sets of reals and real functions.- Ordered sets.- Order types.- Ordinals

This book offers a collection of Set Theory problems and their Solutions. The problems stated are sorted in 31 chapters, the chapters 32 to 62 give the solutions. The book follows a tradition of Hungarian mathematics started with the Polya Szego problem book in Analysis and continued with Lovasz" problem book in Combinatorics. Most of classical set theory is covered, classical in the sense that independence methods are not used, but classical also in the sense, that most results come from the period of 1920-1970. Many problems are also related to other fields of mathematics such as algebra, combinatorics, topology and real analysis.