Volume I

1st Edition - January 1, 1966

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  • Author: K. Kuratowski
  • eBook ISBN: 9781483272566

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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


    § 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

Product details

  • No. of pages: 580
  • Language: English
  • Copyright: © Academic Press 1966
  • Published: January 1, 1966
  • Imprint: Academic Press
  • eBook ISBN: 9781483272566

About the Author

K. Kuratowski

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