Introduction to Set Theory and Topology

Foreword to the First English EditionForeword to the Second English EditionPart I Set Theory Introduction to Part I I. Propositional Calculus § 1. The Disjunction and Conjunction of Propositions § 2. Negation § 3. Implication Exercises II. Algebra of Sets. Finite Operations § 1. Operations on Sets § 2. Inter-Relationship with the Propositional Calculus § 3. Inclusion § 4. Space. Complement of a Set § 5. The axiomatics of the Algebra of Sets § 6. Boolean Algebra. Lattices § 7. Ideals and Filters Exercises III. Propositional Functions. Cartesian Products § 1. The Operation {x: f(x)} § 2. Quantifiers § 3. Ordered Pairs § 4. Cartesian Product § 5. Propositional Functions of Two Variables. Relations § 6. Cartesian Products of n Sets. Propositional Functions of n Variables § 7. On the Axiomatics of Set Theory Exercises IV. The Mapping Concept. Infinite Operations. Families of Sets § 1. The Mapping Concept § 2. Set-Valued Mappings § 3. The Mapping Fx = {y: f(x,y)} § 4. Images and Inverse Images Determined by a Mapping § 5. The Operations ¿R and nR. Covers § 6. Additive and Multiplicative Families of Sets § 7. Borel Families of Sets § 8. Generalized Cartesian Products Exercises V. The Concept of the Power of a Set. Countable Sets § 1. One-to-one Mappings § 2. Power of a Set § 3. Countable Sets Exercises VI. Operations on Cardinal Numbers. The Numbers a and c § 1. Addition and Multiplication § 2. Exponentiation § 3. Inequalities for Cardinal Numbers § 4. Properties of the Number c Exercises VII. Order Relations § 1. Definitions § 2. Similarity. Order Types § 3. Dense Ordering § 4. Continuous Ordering § 5. Inverse Systems, Inverse Limits Exercises VIII. Well Ordering § 1. Well Ordering § 2. Theorem on Transfinite Induction § 3. Theorems on the Comparison of Ordinal Numbers § 4. Sets of Ordinal Numbers § 5. The Number O § 6. The Arithmetic of Ordinal Numbers § 7. The Well-Ordering Theorem § 8. Definitions by Transfinite Induction ExercisesPart II Topology Introduction to Part II IX. Metric Spaces. Euclidean Spaces § 1. Metric Spaces § 2. Diameter of a Set. Bounded Spaces. Bounded Mappings § 3. The Hubert Cube § 4. Convergence of a Sequence of Points § 5. Properties of the Limit § 6. Limit in the Cartesian Product § 7. Uniform Convergence Exercises X. Topological Spaces § 1. Definition. Closure Axioms § 2. Relations to Metric Spaces § 3. Further Algebraic Properties of the Closure Operation § 4. Closed Sets. Open Sets § 5. Operations on Closed Sets and Open Sets § 6. Interior Points. Neighborhoods § 7. The Concept of open Set as the Primitive Term of the Notion of Topological Space § 8. Base and Subbase § 9. Relativization. Subspaces § 10. Comparison of Topologies § 11. Cover of a Space Exercises XI. Basic Topological Concepts § 1. Borel Sets § 2. Dense Sets and boundary Sets § 3. T1-spaces, T2-spaces § 4. Regular Spaces, Normal Spaces § 5. Accumulation Points. Isolated Points § 6. The Derived Set § 7. Sets Dense in Themselves Exercises XII. Continuous Mappings § 1. Continuity § 2. Homeomorphisms § 3. Case of Metric Spaces § 4.