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Foundations of General Topology
1st Edition - January 1, 1964
Author: William J. Pervin
Editor: Ralph P. Boas
9 7 8 - 1 - 4 8 3 2 - 2 5 1 5 - 9
Foundations of General Topology presents the value of careful presentations of proofs and shows the power of abstraction. This book provides a careful treatment of general… Read more
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Foundations of General Topology presents the value of careful presentations of proofs and shows the power of abstraction. This book provides a careful treatment of general topology. Organized into 11 chapters, this book begins with an overview of the important notions about cardinal and ordinal numbers. This text then presents the fundamentals of general topology in logical order processing from the most general case of a topological space to the restrictive case of a complete metric space. Other chapters consider a general method for completing a metric space that is applicable to the rationals and present the sufficient conditions for metrizability. This book discusses as well the study of spaces of real-valued continuous functions. The final chapter deals with uniform continuity of functions, which involves finding a distance that satisfies certain requirements for all points of the space simultaneously. This book is a valuable resource for students and research workers.
PrefaceChapter 1 Algebra of Sets 1.1 Sets and Subsets 1.2 Operations on Sets 1.3 Relations 1.4 Mappings 1.5 Partial OrdersChapter 2 Cardinal and Ordinal Numbers 2.1 Equipotent Sets 2.2 Cardinal Numbers 2.3 Order Types 2.4 Ordinal Numbers 2.5 Axiom of ChoiceChapter 3 Topological Spaces Introduction 3.1 Open Sets and Limit Points 3.2 Closed Sets and Closure 3.3 Operators and Neighborhoods 3.4 Bases and Relative TopologiesChapter 4 Connectedness, Compactness, and Continuity 4.1 Connected Sets and Components 4.2 Compact and Countably Compact Spaces 4.3 Continuous Functions 4.4 Homeomorphisms 4.5 Arcwise ConnectivityChapter 5 Separation and Countability Axioms 5.1 T0- and T1-Spaces 5.2 T2-Spaces and Sequences 5.3 Axioms of Countability 5.4 Separability and Summary 5.5 Regular and Normal Spaces 5.6 Completely Regular SpacesChapter 6 Metric Spaces 6.1 Metric Spaces as Topological Spaces 6.2 Topological Properties 6.3 Hilbert (l2) Space 6.4 Fréchet Space 6.5 Space of Continuous FunctionsChapter 7 Complete Metric Spaces 7.1 Cauchy Sequences 7.2 Completions 7.3 Equivalent Conditions 7.4 Baire TheoremChapter 8 Product Spaces 8.1 Finite Products 8.2 Product Invariant Properties 8.3 Metric Products 8.4 Tichonov Topology 8.5 Tichonov TheoremChapter 9 Function and Quotient Spaces 9.1 Topology of Pointwise Convergence 9.2 Topology of Compact Convergence 9.3 Quotient TopologyChapter 10 Metrization and Paracompactness 10.1 Urysohn's Metrization Theorem 10.2 Paracompact Spaces 10.3 Nagata-Smirnov Metrization TheoremChapter 11 Uniform Spaces 11.1 Quasi Uniformization 11.2 Uniformization 11.3 Uniform Continuity 11.4 Completeness and Compactness 11.5 Proximity SpacesBibliographyIndex