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 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