Book:George McCarty/Topology: An Introduction with Application to Topological Groups
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George McCarty: Topology: An Introduction with Application to Topological Groups
Published $\text {1967}$, Dover Publications, Inc.
- ISBN 0-486-65633-0
Subject Matter
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
Cited by
Errata
Symmetric and Transitive Relation is not necessarily Reflexive: Subset of Cartesian Plane
- Chapter $\text{I}$: Sets and Functions: Relations:
- The shaded regions represent relations
Symmetric and transitive, but not reflexive
Source work progress
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: More About Continuity
- Revisiting from start:
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces
- Exercises after this point not all done
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{M}$