Foundations of General Topology - 1st Edition - ISBN: 9781483200125, 9781483225159

Foundations of General Topology

1st Edition

Authors: William J. Pervin
Editors: Ralph P. Boas
eBook ISBN: 9781483225159
Imprint: Academic Press
Published Date: 1st January 1964
Page Count: 222
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Description

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.

Table of Contents


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

Details

No. of pages:
222
Language:
English
Copyright:
© Academic Press 1964
Published:
Imprint:
Academic Press
eBook ISBN:
9781483225159

About the Author

William J. Pervin

About the Editor

Ralph P. Boas

Ratings and Reviews