# Book:George McCarty/Topology: An Introduction with Application to Topological Groups

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

Published $\text {1967}$, Dover Publications, Inc.

ISBN 0-486-65633-0

### 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
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
Groups of Matrices
Topological Transformation Groups
The Exponential Law: $\paren {Z^Y}^X \cong Z^{X \times Y}$
References and Further Topics
Exercises
Problems
Chapter VIII: THE FUNDAMENTAL GROUP
The Loop Space $\Omega$
The Group $\map {\pi_0} \Omega$
The Fundamental Group $\map {\pi_1} X$
$\map {\pi_1} {R^n}$, A Trivial Example
Further Examples
Homotopies of Maps
Homotopy Types
$\map {\pi_1} {S^n}$, A More Difficult Example
$\map {\pi_1} {X \times Y}$
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

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## Errata

### Symmetric and Transitive Relation is not necessarily Reflexive: Subset of Cartesian Plane

Chapter $\text{I}$: Sets and Functions: Relations: 