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

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: $\paren {Z^Y}^X \cong Z^{X \times Y}$
 * 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



Source work progress
* : $\text{III}$: More About Continuity -- revisiting from start


 * : $\S 5$. Induced mappings; composition; injections; surjections; bijections