Foundations of General Topology

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.

Inhalt

Preface
Chapter 1 Algebra of Sets
1.1 Sets and Subsets
1.2 Operations on Sets
1.3 Relations
1.4 Mappings
1.5 Partial Orders
Chapter 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 Choice
Chapter 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 Topologies
Chapter 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 Connectivity
Chapter 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 Spaces
Chapter 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 Functions
Chapter 7 Complete Metric Spaces
7.1 Cauchy Sequences
7.2 Completions
7.3 Equivalent Conditions
7.4 Baire Theorem
Chapter 8 Product Spaces
8.1 Finite Products
8.2 Product Invariant Properties
8.3 Metric Products
8.4 Tichonov Topology
8.5 Tichonov Theorem
Chapter 9 Function and Quotient Spaces
9.1 Topology of Pointwise Convergence
9.2 Topology of Compact Convergence
9.3 Quotient Topology
Chapter 10 Metrization and Paracompactness
10.1 Urysohn's Metrization Theorem
10.2 Paracompact Spaces
10.3 Nagata-Smirnov Metrization Theorem
Chapter 11 Uniform Spaces
11.1 Quasi Uniformization
11.2 Uniformization
11.3 Uniform Continuity
11.4 Completeness and Compactness
11.5 Proximity Spaces
Bibliography
Index