ProofWiki:Books/George McCarty/Topology: An Introduction with Application to Topological Groups

George McCarty: Topology: An Introduction with Application to Topological Groups

Published 1967, Dover Publications, Inc..

ISBN 0-486-65633-0

Subject Matter

• Topology

Contents

Preface

Introduction

Exercises and Problems

Internal References

Definitions

Set-theoretic Notation

Logic

Special Symbols

Chapter I: SETS AND FUNCTIONS

Unions and Intersections

Relations

Functions

Quotient Functions

Composition of Functions

Factoring Functions

Restrictions and Extensions

References and Further Topics

Exercises

Problems

Chapter II: GROUPS

The Group Property

Subgroups

Morphisms

A Little Number Theory

Quotient Groups

Factoring Morphisms

Direct Products

References and Further Topics

Exercises

Problems

Chapter III: METRIC SPACES

The Definition

$\varepsilon$-balls

Subspaces

A Metric Space of Functions

Pythagoras' Theorem

Path Connectedness

Compactness

$n$-spheres

More About Continuity

References and Further Topics

Exercises

Problems

Chapter IV: TOPOLOGIES

The Definition

Metrizable Spaces and Continuity

Closure, Interior, and Boundary

Subspaces

Bases and Subbases

Product Spaces

Quotient Spaces

Homeomorphisms

Factoring thorough Quotients

References and Further Topics

Exercises

Problems

Chapter V: TOPOLOGICAL GROUPS

The Definition

Homogeneity

Separation

Topological Properties

Coset Spaces

Morphisms

Factoring Morphisms

A Quotient Example

Direct Products

References and Further Topics

Exercises

Problems

Chapter VI COMPACTNESS AND CONNECTEDNESS

Connectedness

Components of Groups

Path Components

Compactness

One-Point Compactification

Regularity and $T_3$ Spaces

Two Applications to Topological Groups

Products

Products of Groups

Products of Spaces

Cross-Sections

Productive Properties

Connected Products

Tychonoff for Two

References and Further Topics

Exercises

Problems

Chapter VII: FUNCTION SPACES

The Definition

Admissible Topologies

Groups of Matrices

Topological Transformation Groups

The Exponential Law: $\left({Z^Y}\right)^X \cong Z^{X \times Y}$

References and Further Topics

Exercises

Problems

Chapter VIII: THE FUNDAMENTAL GROUP

The Loop Space $\Omega$

The Group $\pi_0 \left({\Omega}\right)$

The Fundamental Group $\pi_1 \left({X}\right)$

$\pi_1 \left({R^n}\right)$, A Trivial Example

Further Examples

Homotopies of Maps

Homotopy Types

$\pi_1 \left({S^n}\right)$, A More Difficult Example

$\pi_1 \left({X \times Y}\right)$

References

Exercises

Problems

Chapter IX: THE FUNDAMENTAL GROUP OF THE CIRCLE

The Path Group of a Topological Group

The Universal Covering Group

The Path Group of the Circle

The Universal Covering Group of the Circle

Some Nontrivial Fundamental Groups

The Fundamental Theorem of Algebra

References

Exercises

Problems

Chapter X: LOCALLY ISOMORPHIC GROUPS

The Definition

The Simple Connectivity of $\tilde G$

The Uniqueness of $\tilde G$

The Class for $\R$

References and Further Topics

Exercises

Problems

Greek Alphabet

Symbol Index

Author Index

Subject Index