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Table of Contents

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.