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.
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ISBN-13
978-1-4612-0833-4 (9781461208334)
DOI
10.1007/978-1-4612-0833-4
Schweitzer Klassifikation
1: Metric Spaces.- 1.1 Metric and Pseudo-Metric Spaces.- 1.2 Stone's Theorem.- 1.3 The Metrization Problem.- 1.4 Topology of Metric Spaces.- 1.5 Uniform Continuity and Uniform Convergence.- 1.6 Completeness.- 1.7 Completions.- 2: Uniformities.- 2.1 Covering Uniformities.- 2.2 Uniform Continuity.- 2.3 Uniformizability and Complete Regularity.- 2.4 Normal Coverings.- 3: Transfinite Sequences.- 3.1 Background.- 3.2 Transfinite Sequences in Uniform Spaces.- 3.3 Transfinite Sequences and Topologies.- 4: Completeness, Cofinal Completeness And Uniform Paracompactness.- 4.1 Introduction.- 4.2 Nets.- 4.3 Completeness, Cofinal Completeness and Uniform Paracompactness.- 4.4 The Completion of a Uniform Space.- 4.5 The Cofinal Completion or Uniform Paracompactification.- 5: Fundamental Constructions.- 5.1 Introduction.- 5.2 Limit Uniformities.- 5.3 Subspaces, Sums, Products and Quotients.- 5.4 Hyperspaces.- 5.5 Inverse Limits and Spectra.- 5.6 The Locally Fine Coreflection.- 5.7 Categories and Functors.- 6: Paracompactifications.- 6.1 Introduction.- 6.2 Compactifications.- 6.3 Tamano's Completeness Theorem.- 6.4 Points at Infinity and Tamano's Theorem.- 6.5 Paracompactifications.- 6.6 The Spectrum of ?X.- 6.7 The Tamano-Morita Paracompactification.- 7: Realcompactifications.- 7.1 Introduction.- 7.2 Realcompact Spaces.- 7.3 Realcompactifications.- 7.4 Realcompact Spaces and Lindelöf Spaces.- 7.5 Shirota's Theorem.- 8: Measure And Integration.- 8.1 Introduction.- 8.2 Measure Rings and Algebras.- 8.3 Properties of Measures.- 8.4 Outer Measures.- 8.5 Measurable Functions.- 8.6 The Lebesgue Integral.- 8.7 Negligible Sets.- 8.8 Linear Functional and Integrals.- 9: Haar Measure In Uniform Spaces.- 9.1 Introduction.- 9.2 Haar Integrals and Measures.- 9.3 Topological Groups andUniqueness of Haar Measures.- 10: Uniform Measures.- 10.1 Introduction.- 10.2 Prerings and Loomis Contents.- 10.3 The Haar Functions.- 10.4 Invariance and Uniqueness of Loomis Contents and Haar Measures.- 10.5 Local Compactness and Uniform Measures.- 11: Spaces Of Functions.- 11.1 LP -spaces.- 11.2 The Space L2(?) and Hilbert Spaces.- 11.3 The Space LP(?) and Banach Spaces.- 11.4 Uniform Function Spaces.- 12: Uniform Differentiation.- 12.1 Complex Measures.- 12.2 The Radon-Nikodym Derivative.- 12.3 Decompositions of Measures and Complex Integration.- 12.4 The Riesz Representation Theorem.- 12.5 Uniform Derivatives of Measures.
