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.
Inhalt
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. Fs-Sets, Gd-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 Mappi…