Topology

Topology, Volume I deals with topology and covers topics ranging from operations in logic and set theory to Cartesian products, mappings, and orderings. Cardinal and ordinal numbers are also discussed, along with topological, metric, and complete spaces. Great use is made of closure algebra. Comprised of three chapters, this volume begins with a discussion on general topological spaces as well as their specialized aspects, including regular, completely regular, and normal spaces. Fundamental notions such as base, subbase, cover, and continuous mapping, are considered, together with operations such as the exponential topology and quotient topology. The next chapter is devoted to the study of metric spaces, starting with more general spaces, having the limit as its primitive notion. The space is assumed to be metric separable, and this includes problems of cardinality and dimension. Dimension theory and the theory of Borei sets, Baire functions, and related topics are also discussed. The final chapter is about complete spaces and includes problems of general function theory which can be expressed in topological terms. The book includes two appendices, one on applications of topology to mathematical logics and another to functional analysis. This monograph will be helpful to students and practitioners of algebra and mathematics.

Preface to the First VolumeIntroduction § 1. Operations in Logic and Set Theory I. Algebra of Logic II. Algebra of Sets III. Propositional Functions IV. The Operation E V. Infinite Operations on Sets VI. The Family of All Subsets of a Given Set VII. Ideals and Filters § 2. Cartesian Products I. Definition II. Rules of Cartesian Multiplication III. Axes, Coordinates, and Projections IV. Propositional Functions of Many Variables V. Connections Between the Operators E and V VI. Multiplication by an Axis VII. Relations. The Quotient-Family VIII. Congruence Modulo an Ideal § 3. Mappings. Orderings. Cardinal and Ordinal Numbers I. Terminology and Notation II. Images and Counterimages III. Operations on Images and Counterimages IV. Commutative Diagrams V. Set-Valued Mappings VI. Sets of Equal Power. Cardinal Numbers VII. Characteristic Functions VIII. Generalized Cartesian Products IX. Examples of Countable Products X. Orderings XI. Well Ordering. Ordinal Numbers XII. The Set XNa XIII. Inverse Systems, Inverse Limits XIV. The (A)-operation XV. Lusin Sieve XVI. Application to the Cantor Discontinuum CChapter One Topological Spaces § 4. Definitions. Closure Operation I. Definitions II. Geometrical Interpretation III. Rules of Topological Calculus IV. Relativization V. Logical Analysis of the System of Axioms § 5. Closed Sets, Open Sets I. Definitions II. Operations III. Properties IV. Relativization V. Fσ-Sets, Gδ-Sets VI. Borel Sets VII. Cover of a Space. Refinement VIII. Hausdorff Spaces IX. T 0-Spaces X. Regular Spaces XI. Base and Subbase § 6. Boundary and Interior of a Set I. Definitions II. Formulas III. Relations to Closed and to Open Sets IV. Addition Theorem V. Separated Sets VI. Duality Between the Operations A and A° = Int (A) § 7. Neighbourhood of a Point. Localization of Properties I. Definitions II. Equivalences III. Converging Filters IV. Localization V. Locally Closed Set § 8. Dense Sets, Boundary Sets and Nowhere Dense Sets I. Definitions II. Necessary and Sufficient Conditions III. Operations IV. Decomposition of the Boundary V. Open Sets Modulo Nowhere Dense Sets VI. Relativization VII. Localization VIII. Closed Domains IX. Open Domains § 9. Accumulation Points I. Definitions II. Equivalences III. Formulas IV. Discrete Sets V. Sets Dense in Themselves VI. Scattered Sets § 10. Sets of the First Category (Meager Sets) I. Definition II. Properties III. Union Theorem IV. Relativization V. Localization VI. Decomposition Formulas *§ 11. Open Sets Modulo First Category Sets. Baire Property I. Definition II. General Remarks III. Operations IV. Equivalences IVa. Existence Theorems V. Relativization VI. Baire Property in the Restricted Sense VII. (A)-Operation § 12. Alternated Series of Closed Sets I. Formulas of the General Set Theory II. Definition III. Separation Theorems. Resolution Into Alternating Series IV. Properties of the Remainder V. Necessary and Sufficient Conditions VI. Properties of Resolvable Sets VII. Residues VIII. Residues of Transfinite Order § 13. Continuity. Homeomorphism I. Definition II. Necessary and Sufficient Conditions III. The Set D(f) of Points of Discontinuity IV. Continuous Mappings V. Relativization. Restriction. Retraction VI. Real-Valued Functions. Characteristic Functions VII. One-To-One Continuous Mappings. Comparison of Topologies VIII. Homeomorphism IX. Topological Properties X. Topological Rank XI. Homogeneous Spaces XII. Applications to Topological Groups XIII. Open Mappings. Closed Mappings XIV. Open and Closed Mappings at a Given Point XV. Bicontinuous Mappings § 14. Completely Regular Spaces. Normal Spaces I. Completely Regular Spaces II. Normal Spaces III. Combinatorially Similar Systems of Sets in Normal Spaces IV. Real-Valued Functions Defined on Normal Spaces V. Hereditary Normal Spaces VI. Perfectly Normal Spaces § 15. Cartesian Product X × Y of Topological Spaces I. Definition II. Projections and Continuous Mappings III. Operations on Cartesian Products IV. Diagonal V. Properties of f Considered as Subset of X × Y VI. Horizontal and Vertical Sections. Cylinder on A ⊂ X VII. Invariants of Cartesian Multiplication § 16. Generalized Cartesian Products I. Definition II. Projections and Continuous Mappings III. Operations on Cartesian Products IV. Diagonal V. Invariants of Cartesian Multiplications VI. Inverse Limits § 17. The Space 2X. Exponential Topology I. Definition II. Fundamental Properties III. Continuous Set-Valued Functions IV. Case of X Regular V. Case of X Normal VI. Relations of 2X to Lattices and to Brouwerian Algebras § 18. Semi-Continuous Mappings I. Definitions II. Examples. Relation to Real-Valued Semi-Continuous Functions. Remarks III. Fundamental Properties IV. Union of Semi-Continuous Mappings V. Intersection of Semi-Continuous Mappings VI. Difference of Semi-Continuous Mappings § 19. Decomposition Space. Quotient Topology I. Definition II. Projection. Relationship to Bicontinuous Mappings III. Examples and Remarks IV. Relationship of Quotient Topology to Exponential TopologyChapter Two Metric SpacesA. Relations to Topological Spaces. ℒ*-Spaces § 20. ℒ*-Spaces (provided with the Notion of Limit) I. Definition II. Relation to Topological Spaces III. Notion of Continuity IV. Cartesian Product of ℒ*-Spaces V. Countably Compact ℒ*-Spaces VI. Continuous Convergence. The Set YX as an ℒ-Space VII. Operations on the Spaces YX with ℒ*-Topology VIII. Continuous Convergence in the Narrow Sense IX. Moore-Smith Convergence (Main Definitions) § 21. Metric Spaces. General Properties I. Definitions II· Topology in Metric Spaces III. Diameter. Continuity. Oscillation IV. The Number ρ(A, B). Generalized Ball. Normality of Metric Spaces V. Shrinking Mapping VI. Metrization of the Cartesian Product VII. Distance of Two Sets. The Space (2X)m VIII. Totally Bounded Spaces IX. Equivalence Between Countably Compact Metric Spaces and Compact Metric Spaces X. Uniform Convergence. Metrization of the Space YX XI. Extension of Relatively Closed or Relatively Open Sets XII. Refinements of Infinite Covers XIII. Gδ-Sets in Metric Spaces XIV. Proximity Spaces. Uniform Spaces (Main Definitions) XV. Almost-Metric Spaces XVI. Paracompactness of Metric Spaces XVII. Metrization Problem § 22. Spaces with a Countable Base I. General Properties II. Metrization and Introduction of Coordinates III. Separability of the Space YX IV. Reduction of Closed Sets to Individual Points V. Products of Spaces with a Countable Base. Sets of the First Category *VI. Products of Spaces with a Countable Base. Baire PropertyB. Cardinality Problems § 23. Power of the Space. Condensation Points I. Power of the Space II. Dense Parts III. Condensation Points IV. Properties of the Operation X0 V. Scattered Sets VI. Unions of Scattered Sets VII. Points of Order m VIII. The Concept of Effectiveness § 24. Powers of Various Families of Sets I. Families of Open Sets. Families of Sets with the Baire Property II. Well Ordered Monotone Families III. Resolvable Sets IV. Derived Sets of Order α V. Logical Analysis VI. Families of Continuous Functions VII. Structure of Monotone Families of Closed Sets VIII. Strictly Monotone Families IX. Relations of Strictly Monotone Families to Continuous Functions X. Strictly Monotone Families of Closed Order TypesC. Problems of Dimension § 25. Definitions. General Properties I. Definition of Dimension II. Dimension of Subsets III. The Set E(n) § 26. 0-Dimensional Spaces I. Base of the Space II. Reduction and Separation Theorems III. Union Theorems for 0-Dimensional Sets IV. Extension of 0-Dimensional Sets V. Countable Spaces § 27. n-Dimensional Spaces I. Union Theorems II. Separation of Closed Sets III. Decomposition of an n-Dimensional Space. Condition Dn IV. Extension of n-Dimensional Sets V. Dimensional Kernel VI. Weakly n-Dimensional Spaces VII. Dimensionalizing Families VIII. Dimension of the Cartesian Product IX. Continuous and One-To-One Mappings of n-Dimensional Spaces X. Remarks on the Dimension Theory in Arbitrary Metric Spaces § 28. Simplexes, Complexes, Polyhedra I. Definitions II. Topological Dimension of a Simplex III. Applications to the Fixed Point Problem IV. Applications to the Cubes Jn and JN0 V. Nerve of a System of Sets VI. Mappings of Metric Spaces Into Polyhedra VII. Approximation of Continuous Functions by Means of Kappa Functions VIII. Infinite Complexes and Polyhedra IX. Extension of Continuous FunctionsD. Countable Operations. Borel Sets. B Measurable Functions § 29. Lower and Upper Limits I. Lower Limit II. Formulas III. Upper Limit Iv. Formulas V. Relations Between Li and Ls VI. Limit VII. Relativization VIII. Generalized Bolzano-Weierstrass Theorem IX. The Space (2X)L *§ 30. Borel Sets I. Equivalences II. Classification of Borel Sets III. Properties of the Classes Fα and Gα IV. Ambiguous Borel Sets V. Decomposition of Borel Sets Into Disjoint Sets VI. Alternated Series of Borel Sets VII. Reduction and Separation Theorems VIII. Relatively Ambiguous Sets IX. The Limit Set of Ambiguous Sets X. Locally Borel Sets. Montgomery Operation M XI. Evaluation of Classes with the Aid of Logical Symbols XII. Applications XIII. Universal Functions XIV. Existence of Sets of Class Gα Which Are Not of Class Fα XV. Problem of Effectiveness *§ 31. B Measurable Functions I. Classification II. Equivalences III. Composition of Functions IV. Partial Functions V. Functions of Several Variables VI. Complex and Product Functions VII. Graph of f: X × Y VIII. Limit of Functions IX. Analytic Representation X. Baire Theorems on Functions of Class 1 *§ 32. Functions Which Have the Baire Property I. Definition II. Equivalences III. Operations on Functions Which Have the Baire Property IV. Functions Which Have the Baire Property in the Restricted Sense V. Relations to the Lebesgue MeasureChapter Three Complete Spaces § 33. Definitions. General Properties I. Definitions II. Convergence and Cauchy Sequences III. Cartesian Product IV. The Space (2X)m V. Function Space VI. Complete Metrization of Gδ-Sets VII. Imbedding of a Metric Space in a Complete Space § 34. Sequences of Sets. Baire Theorem I. The Coefficient α(A) II. Cantor Theorem III. Application to Continuous Functions IV. Baire Theorem V. Applications to Gδ-Sets VI. Applications to Fδ and Gδ Sets VII. Application to Functions of Class 1 VIII. Applications to Existence Theorems § 35. Extension of Functions I. Extension of Continuous Functions II. Extension of Homeomorphisms III. Topological Characterization of Complete Spaces IV. Intrinsic Invariance of Various Families of Sets V. Applications to Topological Ranks VI. Extension of B Measurable Functions VII. Extension of a Homeomorphism of Class α, β. *§ 36. Relations of Complete Separable Spaces to the Space N of Irrational Numbers I. Operation (A) II. Mappings of the Space N Onto Complete Spaces III. One-To-One Mappings IV. Decomposition Theorems V. Relations to the Cantor Discontinuum C *§ 37. Borel Sets in Complete Separable Spaces I. Relations of Borel Sets to the Space N II. Characterization of the Borel Class with Aid of Generalized Homoeomorphisms III. Resolution of Ambiguous Sets Into Alternate Series IV. Small Borel Classes *§38. Projective Sets I. Definitions II. Relations Between Projective Classes III. Properties of Projective Sets IV. Projections V. Universal Functions VI. Existence Theorem VII. Invariance VIII. Projective Propositional Functions IX. Invariance of Projective Classes with Respect to the Sieve Operation and the Operation (A) X. Transfinite Induction XI. Hausdorff Operations §39. Analytic Sets I. General Theorems II. Analytic Sets as Results of the Operation (A) III. First Separation Theorem IV. Applications to Borel Sets V. Applications to B Measurable Functions VI. Second Separation Theorem VII. Order of Value of a B Measurable Function VIII. Constituents of CA Sets IX. Projective Class of a Propositional Function that Involves Variable Order Types X. Reduction Theorems XI. A and CA Functions § 40. Totally Imperfect Spaces and Other Singular Spaces I. Totally Imperfect Spaces II. Spaces that Are Always of the First Category III. Rarefied Spaces (or λ Spaces) IV. Mappings *V. Property λ′ VI. σ Spaces VII. v Spaces, Concentrated Spaces, Property C VIII. Relation to the Baire Property in the Restricted Sense IX. Relation of the v Spaces to the General Set TheoryAppendix A. (By A. Mostowski.) Some Applications of Topology to Mathematical Logic I. Classifications of Definable Sets II. The Space of Ideals and the Proof of Completeness of the Logic of Predicates III. Non-Classical Logics IV. Other ApplicationsAppendix B. (By E. Sikorski.) Applications of Topology to Functional AnalysisList of Important SymbolsAuthor IndexSubject Index

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