Theorems and Counterexamples in Mathematics

1 Algebra.- 1.1 Group Theory.- 1.1.1 Axioms.- 1.1.2 Subgroups.- 1.1.3 Exact versus splitting sequences.- 1.1.4 The functional equation:f (x +y) = f (x) + f(y).- 1.1.5 Free groups; free topological groups.- 1.1.6 Finite simple groups.- 1.2 Algebras.- 1.2.1 Division algebras ("noncommutative fields").- 1.2.2 General algebras.- 1.2.3 Miscellany.- 1.3 Linear Algebra.- 1.3.1 Finite-dimensional vector spaces.- 1.3.2 General vector spaces.- 1.3.3 Linear programming.- 2 Analysis.- 2.1 Classical Real Analysis.- 2.1.1 ?X.- 2.1.2 Derivatives and extrema.- 2.1.3 Convergence of sequences and series.- 2.1.4 ?X × Y.- 2.2 Measure Theory.- 2.2.1 Measurable and nonmeasurable sets.- 2.2.2 Measurable and nonmeasurable functions.- 2.2.3 Group-invariant measures.- 2.3 Topological Vector Spaces.- 2.3.1 Bases.- 2.3.2 Dual spaces and reflexivity.- 2.3.3 Special subsets of Banach spaces.- 2.3.4 Function spaces.- 2.4 Topological Algebras.- 2.4.1 Derivations.- 2.4.2 Semisimplicity.- 2.5 Differential Equations.- 2.5.1 Wronskians.- 2.5.2 Existence/uniqueness theorems.- 2.6 Complex Variable Theory.- 2.6.1 Morera's theorem.- 2.6.2 Natural boundaries.- 2.6.3 Square roots.- 2.6.4 Uniform approximation.- 2.6.5 Rouché's theorem.- 2.6.6 Bieberbach's conjecture.- 3 Geometry/Topology.- 3.1 Euclidean Geometry.- 3.1.1 Axioms of Euclidean geometry.- 3.1.2 Topology of the Euclidean plane.- 3.2 Topological Spaces.- 3.2.1 Metric spaces.- 3.2.2 General topological spaces.- 3.3 Exotica in Differential Topology.- 4 Probability Theory.- 4.1 Independence.- 4.2 Stochastic Processes.- 4.3 Transition Matrices.- 5 Foundations.- 5.1 Logic.- 5.2 Set Theory.- Supplemental Bibliography.- Symbol List.- Glossary/Index.