Modern Analysis and Topology

The purpose of this book is to provide an integrated development of modern analysis and topology through the integrating vehicle of uniform spaces. It is intended that the material be accessible to a reader of modest background. An advanced calculus course and an introductory topology course should be adequate. But it is also intended that this book be able to take the reader from that state to the frontiers of modern analysis and topology in-so-far as they can be done within the framework of uniform spaces. Modern analysis is usually developed in the setting of metric spaces although a great deal of harmonic analysis is done on topological groups and much offimctional analysis is done on various topological algebraic structures. All of these spaces are special cases of uniform spaces. Modern topology often involves spaces that are more general than uniform spaces, but the uniform spaces provide a setting general enough to investigate many of the most important ideas in modern topology, including the theories of Stone-Cech compactification, Hewitt Real-compactification and Tamano-Morita Para­ compactification, together with the theory of rings of continuous functions, while at the same time retaining a structure rich enough to support modern analysis.

1: Metric Spaces.- 1.1 Metric and Pseudo-Metric Spaces.- Distance Functions, Spheres, Topology of Pseudo-Metric Spaces, The Ring C*(X), Real Hilbert Space, The Distance from a Point to a Set, Partitions of Unity.- 1.2 Stone’s Theorem.- Refinements, Star Refinements and ?-Refinements, Full Normality, Paracompactness, Shrinkable Coverings, Stone’s Theorem.- 1.3 The Metrization Problem.- Functions That Can Distinguish Points from Sets, ?-Local Finiteness, Urysohn’s Metrization Theorem, The Nagata-Smirnov Metrization Theorem, Local Starrings, Arhangel’skil’s Metrization Theorem.- 1.4 Topology of Metric Spaces.- Complete Normality and Perfect Normality, First and Second Countable Spaces, Separable Spaces, The Diameter of a Set, The Lebesgue Number, Precompact Spaces, Countably Compact and Sequentially Compact Spaces.- 1.5 Uniform Continuity and Uniform Convergence.- Uniform Continuity, Uniform Homeomorphisms and Isomorphisms, Isometric Functions, Uniform Convergence.- 1.6 Completeness.- Convergence and Clustering of Sequences, Cauchy Sequences and Cofinally Cauchy, Sequences, Complete and Cofinally Complete Spaces, The Lebesgue Property, Borel Compactness, Regularly Bounded Metric Spaces.- 1.7 Completions.- The Completion of a Metric Space, Uniformly Continuous Extensions.- 2: Uniformities.- 2.1 Covering Uniformities.- Uniform Spaces, Normal Sequences of Coverings, Bases and Subbases for Uniformities, Normal Coverings, Uniform Topology.- 2.2 Uniform Continuity.- Uniform Continuity, Uniform Homeomorphisms, Pseudo-Metrics Determined by Normal Sequences.- 2.3 Uniformizability and Complete Regularity.- Uniformizable Spaces, The Equivalence of Uniformizability and Complete Regularity, Regularly Open Sets and Coverings, Open and Closed Bases of Uniformities, Regularly Open Bases of Uniformities, Universal or Fine Uniformities.- 2.4 Normal Coverings.- The Unique Uniformity of a Compact Hausdorff Space, Tukey’s Characterization of Normal Spaces, Star-Finite Coverings, Precise Refinements, Some Results of K. Morita, Some Corrections of Tukey’s Theorems by Morita.- 3: Transfinite Sequences.- 3.1 Background.- 3.2 Transfinite Sequences in Uniform Spaces.- Cauchy and Cofinally Cauchy Transfinite Sequences, A Characterization of Paracompactness in Terms of Transfinite Sequences, Shirota’s e Uniformity, Some Characterizations of the Lindelöf Property in Terms of Transfinite Sequences, The ? Uniformity, A Characterization of Compactness in Terms of Transfinite Sequences.- 3.3 Transfinite Sequences and Topologies.- Characterizations of Open and Closed Sets in Terms of Transfinite Sequences, A Characterization of the Hausdorff Property in Terms of Transfinite Sequences, Cluster Classes and the Characterization of Topologies, A Characterization of Continuity in Terms of Transfinite Sequences.- 4: Completeness, Cofinal Completeness And Uniform Paracompactness.- 4.1 Introduction.- 4.2 Nets.- Convergence and Clustering of Nets, Characterizations of Open and Closed Sets in Terms of Nets, A Characterization of the Hausdorff Property in Terms of Nets, Subnets, A Characterization of Compactness in Terms of Nets, A Characterization of Continuity in Terms of Nets, Convergence Classes and the Characterization of Topologies, Universal Nets, Characterizations of Paracompactness, the Lindelöf Property and Compactness in Terms of Nets.- 4.3 Completeness, Cofinal Completeness and Uniform Paracompactness.- Cauchy and Cofinally Cauchy Nets, Completeness and Cofinal Completeness, The Lebesgue Property, Precompactness, Uniform Paracompactness.- 4.4 The Completion of a Uniform Space.- Fundamental Nets, Completeness in Terms of Fundamental Nets, The Construction of the Completion with Fundamental Nets, The Uniqueness of the Completion.- 4.5 The Cofinal Completion or Uniform Paracompactification.- The Topological Completion, Preparacompactness, Countable Bound-edness and the Lindelöf Property, A Necessary and Sufficient Condition for a Uniform Space to Have a Paracompact Completion, A Necessary and Sufficient Condition for a Uniform Space to Have a Lindelöf Completion, The Existence of the Cofinal Completion, A Characterization of Preparacompactness.- 5: Fundamental Constructions.- 5.1 Introduction.- 5.2 Limit Uniformities.- Infimum and Supremum Topologies, Infimum and Supremum Uniformities, Projective and Inductive Limit Topologies, Projective and Inductive Limit Uniformities.- 5.3 Subspaces, Sums, Products and Quotients.- Uniform Product Spaces, Uniform Subspaces, Quotient Uniform Spaces, The Uniform Sum.- 5.4 Hyperspaces.- The Hyperspace of a Uniform Space, Supercompleteness, Burdick’s Characterization of Supercompleteness, Other Characterizations of Supercompleteness, Supercompleteness and Cofinal Completeness, Paracompactness and Supercompleteness.- 5.5 Inverse Limits and Spectra.- Inverse Limit Sequences, Inverse Limit Systems, Inverse Limit Systems of Uniform Spaces, Morita’s Weak Completion, The Spectrum of Weakly Complete Uniform Spaces, Morita’s and Pasynkov’s Characterizations of Closed Subsets of Products of Metric Spaces.- 5.6 The Locally Fine Coreflection.- Uniformly Locally Uniform Coverings, Locally Fine Uniform Spaces, The Derivative of a Uniformity, Partially Cauchy Nets, Injective Uniform Spaces, Subfine Uniform Spaces, The Subfine Coreflection.- 5.7 Categories and Functors.- Concrete Categories, Objects, Morphisms, Covariant Functors, Isomorphisms, Monomorphisms, Duality, Subcategories, Reflection, Coreflection.- 6: Paracompactifications.- 6.1 Introduction.- Some Problems of K. Morita and H. Tamano, Topological Completion, Paracompactifications, Compactifications, Samuel Compactifications, The Stone-?ech Compactification, Uniform Paracompactifications, Tamano’s Paracompactification Problem.- 6.2 Compactifications.- Extensions of Open Sets, Extensions of Coverings, The Extent of a Covering, Stable Coverings, Star-Finite Partitions of Unity.- 6.3 Tamano’s Completeness Theorem.- The Radical of a Uniform Space, Tamano’s Completeness Theorem, Necessary and Sufficient Conditions for Topological Completeness.- 6.4 Points at Infinity and Tamano’s Theorem.- Points and Sets at Infinity, Some Characterizations of Paracompactness by Tamano, Tamano’s Theorem.- 6.5 Paracompactifications.- Completions of Uniform Spaces as Subsets of ?X, A Solution of Tamano’s Paracompactification Problem, The Tamano-Morita Paracompactification, Characterizations of Paracompactness, the Lindelöf Property and Compactness in Terms of Supercompleteness, Another Necessary and Sufficient Condition for a Uniform Space to Have a Paracompact Completion, Another Necessary and Sufficient Condition for a Uniform Space to Have a Lindelöf Completion, The Definition and Existence of the Supercompletion.- 6.6 The Spectrum of ?X.- The Spectrum of ?X, The Spectrum of uX, Monta’s Weak Completion.- 6.7 The Tamano-Morita Paracompactification.- M-spaces, Perfect and Quasi-perfect Mappings, The Topological Completion of an M-space, The Tamano-Morita Paracompactification of an M-space.- 7: Realcompactifications.- 7.1 Introduction.- Another Characterization of ?X, Q-spaces, CZ-maximal Families.- 7.2 Realcompact Spaces.- Realcompact Spaces, The Hewitt Realcompactification, Characterizations of Realcompactness, Properties of Realcompact Spaces, Pseudo-metric Uniformities, The c and c* Uniformities.- 7.3 Realcompactifications.- Realcompactifications, The Equivalence of uX and eX, The Uniqueness of the Hewitt Realcompactification, Characterizations of uX, Properties of uX, Hereditary Realcompactness.- 7.4 Realcompact Spaces and Lindelöf Spaces.- Tamano’s Characterization of Realcompact Spaces, A Necessary and Sufficient Condition for the Realcompactification to be Lindelöf, Tamano’s Characterization of Lindelöf Spaces.- 7.5 Shirota’s Theorem.- Measurable Cardinals, {0,1} Measures, The Relationship of Non-Zero {0,1} Measures and CZ-maximal Families, A Necessary and Sufficient Condition for Discrete Spaces to be Realcompact, Closed Classes of Cardinals, Shirota’s Theorem.- 8: Measure And Integration.- 8.1 Introduction.- Riemann Integration, Lebesgue Integration, Measures, Invariant Integrals.- 8.2 Measure Rings and Algebras.- Rings, Algebras, ?-Rings, ?-Algebras, Borel Sets, Baire Sets, Measures, Measure Rings, Measurable Sets, Measure Algebras, Measure Spaces, Complete Measures, The Completion of a Measure, Borel Measures, Lebesgue Measure, Baire Measures, The Lebesgue Ring, Lebesgue Measurable Sets, Finite Measures, Infinite Measures.- 8.3 Properties of Measures.- Monotone Collections, Continuous from Below, Continuous from Above.- 8.4 Outer Measures.- Hereditary Collections, Outer Measures, Extensions of Measures, ?*-Measurability.- 8.5 Measurable Functions.- Measurable Spaces, Measurable Sets, Measurable Functions, Borel Functions, Limits Superior, Limits Inferior, Point-wise Limits of Functions, Simple Functions, Simple Measurable Functions.- 8.6 The Lebesgue Integral.- Development of the Lebesgue Integral.- 8.7 Negligible Sets.- Negligible Sets, Almost Everywhere, Complete Measures, Completion of a Measure.- 8.8 Linear Functional and Integrals.- Linear Functionals, Positive Linear Functionals, Lower Semi-continuous, Upper Semi-continuous, Outer Regularity, Inner Regularity, Regular Measures, Almost Regular Measures, The Riesz Representation Theorem.- 9: Haar Measure In Uniform Spaces.- 9.1 Introduction.- Isogeneous Uniform Spaces, Isomorphisms, Homogeneous Spaces, Translations, Rotations, Reflections, Haar Integral, Haar Measure.- 9.2 Haar Integrals and Measures.- Development of the Haar integral on Locally Compact Isogeneous Uniform Spaces.- 9.3 Topological Groups and Uniqueness of Haar Measures.- Topological Groups, Abelian Topological Groups, Open at 0, Right Uniformity, Left Uniformity, Right Coset, Left Coset, Quotient of a Topological Group, A Necessary and Sufficient Condition for a Locally Compact Space to Have a Topological Group Structure.- 10: Uniform Measures.- 10.1 Introduction.- Uniform Measures, The Congruence Axiom, Loomis Contents.- 10.2 Prerings and Loomis Contents.- Prerings, Hereditary Open Prerings, Loomis Contents, Uniformly Separated, Left Continuity, Invariant Loomis Contents, Zero-boundary Sets.- 10.3 The Haar Functions.- The Haar Covering Function, The Haar Function, Extension of Loomis Contents to Finitely Additive Measures.- 10.4 Invariance and Uniqueness of Loomis Contents and Haar Measures.- Invariance with Respect to a Uniform Covering, Invariance on Compact Spheres, Development of Loomis Contents on Suitably Restricted Uniform Spaces..- 10.5 Local Compactness and Uniform Measures.- Almost Uniform Measures, Uniform Measures, Jordan Contents, Monotone Sequences of Sets, Monotone Classes, Development of Uniform Measures on Suitably Restricted Uniform Spaces.- 11: Spaces Of Functions.- 11.1 LP -spaces.- Conjugate Exponents, LP-norm, The Essential Supremum, Essentially Bounded, Minkowski’s Inequality Hölder’s Inequality, The Supremum Norm, The Completion of CK(X) with Respect to the LP-norm.- 11.2 The Space L2(?) and Hilbert Spaces.- Square Integrable Functions, Inner Product, Schwarz Inequality, Hilbert Space, Orthogonality, Orthogonal Projections, Linear Combinations, Linear Independence, Span, Basis of a Vector Space, Orthonormal Sets, Orthonormal Bases, Bessel’s Inequality, Riesz-Fischer Theorem, Hilbert Space Isomorphism.- 11.3 The Space LP(?) and Banach Spaces.- Normed Linear Space, Banach Space, Linear Operators, Kernel of a Linear Operator, Bounded Linear Operators, Dual Spaces, Hahn-Banach Theorem, Second Dual Space, Baire’s Category Theorem, Nowhere Dense Sets, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle, Banach-Steinhaus Theorem.- 11.4 Uniform Function Spaces.- Uniformity of Pointwise Convergence, Uniformity of Uniform Convergence, Joint Continuity, Uniformity of Uniform Convergence on Compacta, Topology of Compact Convergence, Compact-Open Topology, Joint Continuity on Compacta, Ascoli Theorem, Equicontinuity.- 12: Uniform Differentiation.- 12.1 Complex Measures.- Complex Measure, Total Variation, Absolute Continuity, Concentration of a Measure on a Subset, Orthogonality of Measures.- 12.2 The Radon-Nikodym Derivative.- Radon-Nikodym Derivative and its Applications.- 12.3 Decompositions of Measures and Complex Integration.- Polar Decomposition, Lebesgue Decomposition, Complex Integration.- 12.4 The Riesz Representation Theorem.- Regular and Almost Regular Complex Measures, The Riesz Representation Theorem.- 12.5 Uniform Derivatives of Measures.- Differentiation of a Measure at a Point, Differentiable Measures, L1-differentiable Measures, Uniformly Differentiable Measures, Fubini’s Theorem.