Topology
1st Edition
Volume I
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Description
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.
Table of Contents
Preface to the First Volume
Introduction
§ 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 C
Chapter 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 Topology
Chapter Two Metric Spaces
A. 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 Property
B. 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 Types
C. 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 Functions
D. 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 Measure
Chapter 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 Theory
Appendix 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 Applications
Appendix B. (By E. Sikorski.) Applications of Topology to Functional Analysis
List of Important Symbols
Author Index
Subject Index
Details
- No. of pages:
- 580
- Language:
- English
- Copyright:
- © Academic Press 1966
- Published:
- 1st January 1966
- Imprint:
- Academic Press
- eBook ISBN:
- 9781483272566
About the Author
K. Kuratowski
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