Chapter 1 -- Definitions and Elementary Examples -- Actions, Subgroups, Quotient Spaces -- Products of Compact Groups -- Applications to Abelian Groups -- Projective Limits -- Totally Disconnected Compact Groups -- Some Duality Theory -- Postscript -- References for this Chapter-Additional Reading -- Chapter 2 -- Some Basic Representation Theory for Compact Groups -- The Haar Integral -- Consequences of Haar Measure -- The Main Theorem on Hilbert Modules for Compact Groups -- Postscript -- References for this Chapter-Additional Reading -- Chapter 3 -- Part 1: The Classical Theorem of Peter and Weyl -- An Excursion into Linear Algebra -- The G-Modules -- The Fine Structure of R( -- Part 2: The General Theory of G-Modules -- Vector Valued Integration -- The First Application: The Averaging Operator -- Compact Groups Acting on Convex Cones -- More Module Actions, Convolutions -- Complexification of Real Representations -- Postscript -- References for this Chapter-Additional Reading -- Chapter 4 -- Part 1: Characters of Finite Dimensional Representations -- Part 2: The Structure Theorem of -- Cyclic Modules -- Postscript -- References for this Chapter-Additional Reading -- Chapter 5 -- Preliminaries -- The Exponential Function and the Logarithm -- Differentiating the Exponential Function in a Banach Algebra -- Local Groups for the Campbell-Hausdorff Multiplication -- Subgroups of A− -- and Linear Lie Groups -- Analytic Groups -- The Intrinsic Exponential Function of a Linear Lie Group -- The Adjoint Representation of a Linear Lie Group -- Subalgebras, Ideals, Lie Subgroups, Normal Lie Subgroups -- Normalizers, Centralizers, Centers -- The Commutator Subgroup -- Forced Continuity of Morphisms between Lie Groups -- Quotients of Linear Lie Groups -- The Topological Splitting Theorem for Normal Vector Subgroups -- Postscript.

References for this Chapter-Additional Reading -- Chapter 6 -- Compact Lie algebras -- The Commutator Subgroup of a Compact Lie Group -- The Structure Theorem for Compact Lie Groups -- Maximal Tori -- The Second Structure Theorem for Connected Compact Lie Groups -- Compact Abelian Lie Groups and Their Linear Actions -- Action of a Maximal Torus on the Lie Algebra -- The Weyl Group Revisited -- The Commutator Subgroup of Connected Compact Lie Groups -- On the Automorphism Group of a Compact Lie Group -- Covering Groups of Disconnected Compact Lie Groups -- Auerbach's Generation Theorem -- The Topology of Connected Compact Lie Groups -- Postscript -- References for this Chapter-Additional Reading -- Chapter 7 -- The Compact Open Topology and Hom-Groups -- Local Compactness and Duality of Abelian Topological Groups -- Basic Functorial Aspects of Duality -- The Annihilator Mechanism -- Character Groups of Topological Vector Spaces -- The Exponential Function -- Weil's Lemma and Compactly Generated Abelian Groups -- Reducing Locally Compact Groups to Compact Abelian Groups -- The Duality Theorem -- The Identity Component -- The Weight of Locally Compact Abelian Groups -- Postscript -- References for this Chapter-Additional Reading -- Chapter 8 -- Part 1: Aspects of the Algebraic Structurer Divisibility, Torsion, Connectivity -- Compact Abelian Groups as Factor Groups -- Part 2: Aspects of the Point Set Topological Structure Topological Dimension of Compact Abelian Groups -- Arc Connectivity -- Local Connectivity -- Compact Metric Abelian Groups -- Part 3: Aspects of Algebraic Topology-Homotopy Free Compact Abelian Groups -- Homotopy of Compact Abelian Groups -- Exponential Function and Homotopy -- The Fine Structure of Free Compact Abelian Groups -- Part 4: Aspects of Homological Algebra Injective, Projective, and Free Compact Abelian Groups.

Part 5: Aspects of Algebraic Topology-Cohomology Cohomology of Compact Abelian Groups -- Part 6: Aspects of Set Theory Arc Components and Borel Sets -- Postscript -- References for this Chapter-Additional Reading -- Chapter 9 -- Part 1: The Fundamental Structure Theorems of Compact Groups Approximating Compact Groups by Compact Lie Groups -- The Closedness of Commutator Subgroups -- Semisimple Compact Connected Groups -- The Levi-Mal'cev Structure Theorem for Compact Groups -- Maximal Connected Abelian Subgroups -- The Splitting Structure Theorem -- Supplementing the Identity Component -- Part 2: The Structure Theorems for the Exponential Function The Exponential Function of Compact Groups -- The Dimension of Compact Groups -- Locally Euclidean Compact Groups Are Compact Lie Groups -- Part 3: The Connectivity Structure of Compact Groups Arc Connectivity -- Local Connectivity -- Compact Groups and Indecomposable Continua -- Part 4: Some Homological Algebra for Compact Groups The Projective Cover of Connected Compact Groups -- Part 5: The Automorphism Group of Compact Groups -- The Iwasawa Theory of Automorphism Groups -- Simple Compact Groups and the Countable Layer Theorem -- Postscript -- References for this Chapter-Additional Reading -- Chapter 10 -- A Preparation Involving Compact Semigroups -- Orbits, Orbit Space, and Isotropy -- Equivariance and Cross Sections -- The Triviality Theorem -- Quotient Actions, Totally Disconnected G-Spaces -- Compact Lie Groups Acting on Locally Compact Spaces -- Triviality Theorems for Compact Group Actions -- Split Morphisms -- Postscript -- References for this Chapter-Additional Reading -- Chapter 11 -- The Category Theoretical Background -- Splitting the Identity Component -- The Center of a Free Compact Group -- The Commutator Subgroup of a Free Compact Group -- Freeness Versus Projectivity -- Postscript.

References for this Chapter-Additional Reading -- Chapter 12 -- Suitable sets -- Generating Degree and Density -- The Cardinal Invariants of Connected Compact Groups -- Cardinal Invariants in the Absence of Connectivity -- On the Location of Special Generating Sets -- Postscript -- References for this Chapter-Additional Reading -- Appendix 1 -- Examples -- Free Abelian Groups -- Projective Groups -- Torsion Subgroups -- Pure Subgroups -- Free Quotients -- Divisibility -- Some Homological Algebra -- Exact Sequences -- Whitehead's Problem -- Postscript -- References for this Chapter-Additional Reading -- Appendix 2 -- Covering Spaces and Simple Connectivity -- The Group of Covering Transformations -- Universal Covering Groups -- Groups Generated by Local Groups -- Postscript -- References for this Chapter-Additional Reading -- Appendix 3 -- Categories, Morphisms -- Pointed Categories -- Types of Morphisms -- Functors -- Natural Transformations -- Equivalence of Categories -- Limits -- The Continuity of Adjoints -- The Left Adjoint Existence Theorem -- Commutative Monoidal Categories and its Monoid Objects -- Part 1: The Quintessential Diagram Chase -- Part 2: Connected Graded Commutative Hopf Algebras -- Part 3: Duality of Graded Hopf Algebras -- Part 4: An Application to Compact Monoids -- Postscript -- References for this Chapter-Additional Reading -- Appendix 4 -- The Arc Component Topology -- The Weight of a Topological Space -- Metrizability of Topological Groups -- Duality of Vector Spaces -- Subgroups of Topological Groups -- Wallace's Lemma -- Cantor Cubes and Dyadic spaces -- Postscript -- References for this Chapter-Additional Reading.