Alphabeta Math

Math Library

Definitions, theorems, and proofs

For the skeptical readerRich provenance and phase-stratified proofs give you the means to check every claim and every step. Skeptical reading is welcome and encouraged, it's the only path to true understanding.
DashboardYour goals, bookmarks and AI proof help, attached to your Google account and available on every device. Set a theorem as a goal and the library lays out everything it depends on, in reading order.
Proof workflowThe same pipeline on one theorem: a draft, the phase stratification, a judge's rejection, the adjudication, the repair, and the rejudge that closes it.
Editorial workflowThe eleven steps behind every page, from computing the frontier to the single owner pause — with every gate, ledger and refusal the repository actually enforces.
Rabbit holes · Part 1Why is the area of a circle πr²?The familiar formula only means something once area itself is defined. Three axioms pin it down, and the integral makes them precise.

Recent Updates

All 29 from the past 7 days →

In the past 7 days: 27 new pages · 2 pages revised · 1040 results added

Abstract Algebra

Reading order →

78 pages · 39 with a worked companion

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Binary Operations, Monoids, Groups and SubgroupsExamples & counterexamples →0
Cosets, Index and Lagrange's TheoremExamples & counterexamples →1
Rings, Subrings, Integral Domains and FieldsExamples & counterexamples →1
Normal Subgroups and Quotient GroupsExamples & counterexamples →2
Group Homomorphisms and the Isomorphism TheoremsExamples & counterexamples →3
Cyclic Groups and Direct ProductsExamples & counterexamples →4
Free Groups and PresentationsExamples & counterexamples →4
Group Actions, Orbits, Stabilisers and Cayley's TheoremExamples & counterexamples →4
Ideals, Quotient Rings and the Isomorphism Theorems for RingsExamples & counterexamples →4
Divisibility, Euclidean Domains, Principal Ideal Domains and Unique FactorisationExamples & counterexamples →5
Free Products and AmalgamationExamples & counterexamples →5
Modules, Submodules, Quotient Modules and the Isomorphism TheoremsExamples & counterexamples →5
Symmetric Groups, Cycle Decomposition and the Sign HomomorphismExamples & counterexamples →5
The Fundamental Theorem of Finite Abelian GroupsExamples & counterexamples →5
Conjugacy in Sₙ, Generation, and the Simplicity of AₙExamples & counterexamples →6
Free Modules, Exact Sequences, Projective and Injective ModulesExamples & counterexamples →6
Polynomial Rings, the Division Algorithm and RootsExamples & counterexamples →6
Composition Series, the Jordan–Hölder Theorem and Solvable GroupsExamples & counterexamples →7
Simple Field Extensions and the Construction of the Complex NumbersExamples & counterexamples →7
Tensor Products of ModulesExamples & counterexamples →7
The Field of Fractions and LocalisationExamples & counterexamples →7
Chain Conditions, Semisimple Modules and the Wedderburn–Artin TheoremExamples & counterexamples →8
Semidirect Products, Automorphism Groups and Split ExtensionsExamples & counterexamples →8
Splitting FieldsExamples & counterexamples →8
Algebraic Extensions, Extension Degree, and Finite FieldsExamples & counterexamples →9
Modules over a Principal Ideal Domain and the Canonical FormsExamples & counterexamples →9
Sylow's Theorems, p-Groups and Nilpotent GroupsExamples & counterexamples →9
Symmetric Polynomials and the Fundamental Theorem of Symmetric FunctionsExamples & counterexamples →9
Algebraic Closure, Embeddings, and SeparabilityExamples & counterexamples →10
Frattini Subgroups and the Burnside Basis TheoremExamples & counterexamples →10
The Galois CorrespondenceExamples & counterexamples →11
Finite Fields and Cyclotomic ExtensionsExamples & counterexamples →12
The Fundamental Theorem of AlgebraExamples & counterexamples →12
Solvability by Radicals and Kummer TheoryExamples & counterexamples →13
The Group Algebra and Representations of Finite GroupsExamples & counterexamples →13
Finite Averaging and Character-Theory PrerequisitesExamples & counterexamples →14
Maschke's Theorem, Complete Reducibility and the Structure of k[G]Examples & counterexamples →14
Characters and the Orthogonality RelationsExamples & counterexamples →15
Induced Representations, Frobenius Reciprocity and ApplicationsExamples & counterexamples →16

Algebraic Geometry

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12 pages · 6 with a worked companion

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Affine Algebraic Sets and Coordinate RingsExamples & counterexamples →0
Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational MapsExamples & counterexamples →0
Morphisms Local Rings and Rational Maps of Affine VarietiesExamples & counterexamples →1
Projective Algebraic Sets Projective Morphisms and ConesExamples & counterexamples →2
Products Segre and Veronese Embeddings and GrassmanniansExamples & counterexamples →3
Dimension Constructible Images and Dimensions of FibresExamples & counterexamples →4

Algebraic Topology

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46 pages · 23 with a worked companion

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Simplicial Complexes and Simplicial HomologyExamples & counterexamples →0
Simplicial Subdivision and Simplicial ApproximationExamples & counterexamples →1
Singular Chains and Singular HomologyExamples & counterexamples →1
Relative Homology Excision and Mayer VietorisExamples & counterexamples →2
Cw Complexes and Cellular HomologyExamples & counterexamples →3
Homology Axioms Degree and Classical ApplicationsExamples & counterexamples →4
Higher Homotopy Groups and Cofiber SequencesExamples & counterexamples →5
Singular Cohomology and Coefficient TheoremsExamples & counterexamples →5
Cup Cap Cross Products and Cohomology RingsExamples & counterexamples →6
Fibrations Fiber Bundles and Homotopy Exact SequencesExamples & counterexamples →6
Hurewicz Whitehead Freudenthal and Cw ApproximationExamples & counterexamples →7
Orientations Poincare Lefschetz and Alexander DualityExamples & counterexamples →7
Bocksteins Steenrod Squares and Cohomology OperationsExamples & counterexamples →8
Local Coefficients, Twisted Homology, and DualityExamples & counterexamples →8
Spectra and Stable Homotopy GroupsExamples & counterexamples →8
Obstruction Theory, Postnikov Towers, and Classifying SpacesExamples & counterexamples →9
The Serre Spectral Sequence and ApplicationsExamples & counterexamples →10
Topological Vector Bundles and Grassmannian ClassificationExamples & counterexamples →10
Complex Topological K Theory and Bott PeriodicityExamples & counterexamples →11
Leray–Hirsch, the Thom Isomorphism, and Gysin SequencesExamples & counterexamples →11
Generalized Cohomology and the Atiyah Hirzebruch Spectral SequenceExamples & counterexamples →12
Stiefel Whitney and Euler Classes by Universal ConstructionsExamples & counterexamples →12
Chern and Pontryagin Classes by Splitting and ComplexificationExamples & counterexamples →13

Category Theory

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42 pages · 21 with a worked companion

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Categories, Functors and Natural TransformationsExamples & counterexamples →0
Universal Properties, Representables and the Yoneda LemmaExamples & counterexamples →1
Limits and ColimitsExamples & counterexamples →2
Adjunctions Units and CounitsExamples & counterexamples →3
Ends Coends and Weighted LimitsExamples & counterexamples →4
Preadditive and Additive Categories and BiproductsExamples & counterexamples →4
Reflective Subcategories and the Adjoint Functor TheoremsExamples & counterexamples →4
Abelian CategoriesExamples & counterexamples →5
Monads Comonads and Their AlgebrasExamples & counterexamples →5
Kan Extensions Density and the Free CocompletionExamples & counterexamples →6
Monadicity and Beck's TheoremExamples & counterexamples →6
Monoidal Categories and Monoidal FunctorsExamples & counterexamples →6
Subobject Lattices Generators and the Grothendieck AxiomsExamples & counterexamples →6
Closed Monoidal Categories and the Internal HomExamples & counterexamples →7
Exactness and the Member CalculusExamples & counterexamples →7
Strictification and Mac Lanes Coherence TheoremExamples & counterexamples →7
Braided and Symmetric Monoidal CategoriesExamples & counterexamples →8
The Diagram Lemmas in an Abelian CategoryExamples & counterexamples →8
Duality and Rigidity in Monoidal CategoriesExamples & counterexamples →9
Enriched CategoriesExamples & counterexamples →9
Tensor and Fusion CategoriesExamples & counterexamples →10

Combinatorics

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100 pages · 50 with a worked companion

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Finite Counting, Factorials and Binomial CoefficientsExamples & counterexamples →0
Finite Probability Spaces and Random VariablesExamples & counterexamples →1
Formal Power SeriesExamples & counterexamples →1
Inclusion–Exclusion, the Pigeonhole Principle and Double CountingExamples & counterexamples →1
Chains, Antichains, Sperner and DilworthExamples & counterexamples →2
Combinatorial Classes and the Symbolic MethodExamples & counterexamples →2
Graphs, Walks and ConnectivityExamples & counterexamples →2
Linear Recurrences and Rational Generating FunctionsExamples & counterexamples →2
Permutation Statistics, Inversions and Eulerian NumbersExamples & counterexamples →2
Counting up to Symmetry: Burnside and PólyaExamples & counterexamples →3
Eulerian and Hamiltonian GraphsExamples & counterexamples →3
Graph ColouringExamples & counterexamples →3
Incidence Algebras and Möbius InversionExamples & counterexamples →3
Induced Subgraphs and Hereditary Graph ClassesExamples & counterexamples →3
Lattice Paths and Catalan NumbersExamples & counterexamples →3
Linear Algebra Methods in CombinatoricsExamples & counterexamples →3
Ramsey TheoryExamples & counterexamples →3
Set Partitions, Stirling Numbers and Exponential Generating FunctionsExamples & counterexamples →3
Trees, Forests and Spanning TreesExamples & counterexamples →3
Algebraic and Spectral Graph TheoryExamples & counterexamples →4
Block Designs and Finite Projective PlanesExamples & counterexamples →4
Extremal Graph TheoryExamples & counterexamples →4
Integer Partitions and the Twelvefold WayExamples & counterexamples →4
Matchings, Covers, Menger and Network FlowsExamples & counterexamples →4
Finite Probability and the Probabilistic MethodExamples & counterexamples →5
Plane Graphs, Euler's Formula and the Five Colour TheoremExamples & counterexamples →5
Regular Pairs and Induced CountingExamples & counterexamples →6
The Erdős–Hajnal Property and Homogeneous SetsExamples & counterexamples →6
Modules, Substitution and Prime GraphsExamples & counterexamples →7
Sparse Restricted Subgraphs and the Rödl–Nikiforov TheoremsExamples & counterexamples →7
Blockades, Combs and Pattern GraphsExamples & counterexamples →8
Bull-Free Graphs and the Erdős-Hajnal PropertyExamples & counterexamples →8
Classical and Log-Log Erdős–Hajnal BoundsExamples & counterexamples →8
Polynomial Rödl, Virality and Erdős–Hajnal EquivalenceExamples & counterexamples →8
Pure Pairs, Forests and Path–Antipath ClassesExamples & counterexamples →8
Cographs, Perfect Patterns and Pure PairsExamples & counterexamples →9
Quantitative Induced Density and the Log-Log StepExamples & counterexamples →9
Quotient Blockades and Mixing RelationsExamples & counterexamples →9
The Five-Cycle and the Erdős-Hajnal PropertyExamples & counterexamples →10
Iterative Sparsification and the Five-Vertex PathExamples & counterexamples →11
Star Expansions and the Erdős-Hajnal PropertyExamples & counterexamples →11
Iterative Restriction and Comb-Extraction LemmasExamples & counterexamples →12
Small-Graph Erdős-Hajnal ConsequencesExamples & counterexamples →12
Leaf Reducibility and Wonderful FamiliesExamples & counterexamples →13
Comb Structure in co-Bird-Free GraphsExamples & counterexamples →14
Generalized Niceness and Reduction OutcomesExamples & counterexamples →14
From Generalized Niceness to Erdős-HajnalExamples & counterexamples →15
Property (*) and Comb OutcomesExamples & counterexamples →15
The Structural Criterion for Property (*)Examples & counterexamples →16
Comb Structure in co-E-Free GraphsExamples & counterexamples →17

Commutative Algebra

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38 pages · 19 with a worked companion

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Noetherian Rings and Hilbert BasisExamples & counterexamples →0
Localisation of Modules and SupportExamples & counterexamples →1
Prime Spectra and RadicalsExamples & counterexamples →1
Artinian Rings and LengthExamples & counterexamples →2
Associated Primes and Primary DecompositionExamples & counterexamples →2
Flatness and Faithful FlatnessExamples & counterexamples →2
Integral Extensions and Going UpExamples & counterexamples →2
Koszul Complexes and Regular SequencesExamples & counterexamples →2
Noether Normalisation and NullstellensatzExamples & counterexamples →3
Valuation Rings and Discrete Valuation RingsExamples & counterexamples →3
Dedekind Domains and Ideal ClassesExamples & counterexamples →4
Krull Dimension and Height TheoremsExamples & counterexamples →4
Zariski Topology on Prime SpectraExamples & counterexamples →4
Rees Modules Artin Rees and Hilbert Samuel TheoryExamples & counterexamples →5
Inverse Limits and Noetherian CompletionExamples & counterexamples →6
Koszul Euler Characteristics and Hilbert–Samuel MultiplicityExamples & counterexamples →6
Depth and Cohen Macaulay ModulesExamples & counterexamples →7
Henselian Rings and Equicharacteristic Cohen StructureExamples & counterexamples →7
Regular Local Rings and Homological DimensionExamples & counterexamples →8

Complex Analysis

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54 pages · 27 with a worked companion

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Complex Differentiability and the Cauchy–Riemann EquationsExamples & counterexamples →0
Complex Power Series and Analytic FunctionsExamples & counterexamples →1
Contour IntegrationExamples & counterexamples →1
Goursat's Theorem and Cauchy's Theorem in a Convex DomainExamples & counterexamples →2
Analyticity of Holomorphic Functions; Liouville and MoreraExamples & counterexamples →3
The Identity Theorem, the Maximum Principle and the Open Mapping TheoremExamples & counterexamples →4
The Winding Number and the Global Cauchy TheoremExamples & counterexamples →5
Holomorphic Functions of Several Complex VariablesExamples & counterexamples →6
Isolated Singularities and Laurent SeriesExamples & counterexamples →6
Harmonic Functions and the Poisson IntegralExamples & counterexamples →7
Infinite Products and the Weierstrass Factorisation TheoremExamples & counterexamples →7
The Residue Theorem and the Evaluation of Real IntegralsExamples & counterexamples →7
Mittag-Leffler and Runge's TheoremExamples & counterexamples →8
Subharmonic Functions and the Dirichlet ProblemExamples & counterexamples →8
The Argument Principle and Rouché's TheoremExamples & counterexamples →8
The Gamma FunctionExamples & counterexamples →8
The Hartogs PhenomenaExamples & counterexamples →9
The Holomorphic Inverse Function Theorem and Weierstrass PreparationExamples & counterexamples →9
The Riemann Sphere and Möbius TransformationsExamples & counterexamples →9
The Riemann Zeta FunctionExamples & counterexamples →9
Conformal Mapping, Branches, and the Schwarz LemmaExamples & counterexamples →10
Domains of Holomorphy, Plurisubharmonicity and PseudoconvexityExamples & counterexamples →10
Normal Families and Montel's TheoremExamples & counterexamples →10
The Riemann Mapping TheoremExamples & counterexamples →11
Bloch, Schottky, and the Picard TheoremsExamples & counterexamples →12
Simply Connected Plane Domains: the Grand EquivalenceExamples & counterexamples →12
Analytic Continuation, Monodromy, and Riemann SurfacesExamples & counterexamples →13

Computability Theory

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78 pages · 39 with a worked companion

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Expander Graphs and Constraint GraphsExamples & counterexamples →0
Formal Languages, Encodings, and Decision ProblemsExamples & counterexamples →0
Context-Free Grammars and Normal FormsExamples & counterexamples →1
Deterministic Finite Automata and Regular LanguagesExamples & counterexamples →1
Turing Machines, Configurations, and ComputationExamples & counterexamples →1
Turing-Machine Configuration Boundary InterfaceExamples & counterexamples →1
Finite-Automata Transition and Determinisation InterfacesExamples & counterexamples →2
Myhill--Nerode Theory and DFA MinimizationExamples & counterexamples →2
Nondeterministic Finite Automata and Subset ConstructionExamples & counterexamples →2
One-Tape Simulation Normal FormsExamples & counterexamples →2
Pushdown Automata and Context-Free LanguagesExamples & counterexamples →2
Robust Machine Models and Universal ComputationExamples & counterexamples →2
Context-Free Pumping, Ogden's Lemma, and ParsingExamples & counterexamples →3
Decidable, Recognizable, and Enumerable LanguagesExamples & counterexamples →3
Effective Numberings, Reductions, Resources, and RandomnessExamples & counterexamples →3
Primitive Recursive and Partial Computable FunctionsExamples & counterexamples →3
Regular Expressions and Kleene's TheoremExamples & counterexamples →3
Resource Bounds and Machine InvarianceExamples & counterexamples →3
Acceptable Numberings, the s-m-n Theorem, and RecursionExamples & counterexamples →4
Diagonalization and the Halting ProblemExamples & counterexamples →4
Nondeterministic Recursive Padding and Time SeparationExamples & counterexamples →4
P, NP, coNP, and Polynomial ReductionsExamples & counterexamples →4
Randomized Complexity and AmplificationExamples & counterexamples →4
SAT Tableaux and Quantified-Space CompletenessExamples & counterexamples →4
Computable Reductions and Rice's TheoremExamples & counterexamples →5
Interactive Proof Systems and Public CoinsExamples & counterexamples →5
Kolmogorov Complexity and Algorithmic RandomnessExamples & counterexamples →5
Space Complexity, Savitch's Theorem, and TQBFExamples & counterexamples →5
The Cook--Levin TheoremExamples & counterexamples →5
Arithmetization and the Sum-Check ProtocolExamples & counterexamples →6
Classical NP-Completeness ReductionsExamples & counterexamples →6
Counting Complexity and Sharp-PExamples & counterexamples →6
Post Correspondence and Language UndecidabilityExamples & counterexamples →6
The Arithmetical Hierarchy and Post's TheoremExamples & counterexamples →6
Time and Space Hierarchy TheoremsExamples & counterexamples →6
Logarithmic Space, NL, and ReachabilityExamples & counterexamples →7
Oracle Computability, the Jump, and Turing DegreesExamples & counterexamples →7
Boolean Circuits and Nonuniform ComplexityExamples & counterexamples →8
The Polynomial Hierarchy and RelativizationExamples & counterexamples →9

Differential Geometry

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70 pages · 35 with a worked companion

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Euclidean Ordinary Differential Equations with Smooth DependenceExamples & counterexamples →0
Lie Algebra Representations, Enveloping Algebras, and PBWExamples & counterexamples →0
Smooth Manifolds and Smooth MapsExamples & counterexamples →0
Smooth Partitions of Unity and ExhaustionsExamples & counterexamples →1
Solvable and Nilpotent Lie AlgebrasExamples & counterexamples →1
Tangent Cotangent and the DifferentialExamples & counterexamples →1
Rank Theorems and Embedded SubmanifoldsExamples & counterexamples →2
Smooth Vector Bundles and SectionsExamples & counterexamples →3
Sard Theorem and TransversalityExamples & counterexamples →4
Tensor Fields Exterior Algebra and Differential FormsExamples & counterexamples →4
Whitney Embedding Tubular Neighbourhoods and ApproximationExamples & counterexamples →5
Vector Fields Flows and Lie DerivativesExamples & counterexamples →6
Distributions Integral Manifolds and the Frobenius TheoremExamples & counterexamples →7
Manifolds with Boundary Collars and OrientationsExamples & counterexamples →7
The Exterior Derivative and Cartan CalculusExamples & counterexamples →8
Integration of Forms and the General Stokes TheoremExamples & counterexamples →9
Lie Groups, Invariant Fields, and the Exponential MapExamples & counterexamples →9
Lie Subgroups, Actions, and Homogeneous SpacesExamples & counterexamples →10
Measurable Densities and Radon Volume on ManifoldsExamples & counterexamples →10
The De Rham Complex Homotopy and Mayer VietorisExamples & counterexamples →10
Riemannian Metrics Length Distance and VolumeExamples & counterexamples →11
Semisimple Lie Algebras, Cohomology, and Levi TheoryExamples & counterexamples →11
Singular Cochains Mayer Vietoris and Smooth Singular ComparisonExamples & counterexamples →11
Cartan Subalgebras and Root Space DecompositionsExamples & counterexamples →12
Connections Levi Civita and Parallel TransportExamples & counterexamples →12
The de Rham Theorem and DegreeExamples & counterexamples →12
Geodesics, the Exponential Map, Completeness, and Hopf–RinowExamples & counterexamples →13
Root Systems, Dynkin Diagrams, and the Cartan-Killing ClassificationExamples & counterexamples →13
Highest Weight Theory for Complex Semisimple Lie AlgebrasExamples & counterexamples →14
Riemann Curvature and Riemannian SubmanifoldsExamples & counterexamples →14
Symplectic Manifolds, Moser Stability, and Darboux–Weinstein TheoryExamples & counterexamples →14
Compact Lie Groups, Maximal Tori, and Peter–Weyl TheoryExamples & counterexamples →15
Hamiltonian Mechanics and Completely Integrable SystemsExamples & counterexamples →15
Moment Maps and Symplectic ReductionExamples & counterexamples →16
Real Forms and Real Semisimple Lie AlgebrasExamples & counterexamples →16

Differential Topology

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10 pages · 5 with a worked companion

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Morse Critical Points Hessians and IndicesExamples & counterexamples →0
Gradient Like Vector Fields and Morse TrajectoriesExamples & counterexamples →1
Morse Functions Critical Values and GenericityExamples & counterexamples →1
Stable Unstable Manifolds and Morse Smale TransversalityExamples & counterexamples →2
Sublevel Deformation and the Handle Attachment TheoremExamples & counterexamples →2

Fourier Analysis

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14 pages · 7 with a worked companion

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Dirichlet Kernel Localisation and Pointwise Fourier ConvergenceExamples & counterexamples →0
Fejer and Poisson Summability of Fourier SeriesExamples & counterexamples →1
Absolute Convergence and the Wiener AlgebraExamples & counterexamples →2
Divergence and Almost Everywhere Convergence of Fourier SeriesExamples & counterexamples →2
Kolmogorov’s Block Construction and Almost-Everywhere DivergenceExamples & counterexamples →2
Lacunary Fourier Series and Sidon SetsExamples & counterexamples →2
The Carleson–Hunt Time–Frequency TheoremExamples & counterexamples →2

Functional Analysis

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60 pages · 30 with a worked companion

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Fourier Transform Convolution and Approximate IdentitiesExamples & counterexamples →0
Normed and Banach SpacesExamples & counterexamples →0
Bounded Linear Operators and Quotient SpacesExamples & counterexamples →1
Finite Dimensional Normed Spaces and Riesz LemmaExamples & counterexamples →2
Schwartz Space and the Plancherel TheoremExamples & counterexamples →2
Sequential Uniform Boundedness with Countable ChoiceExamples & counterexamples →2
The Analytic Hahn Banach TheoremExamples & counterexamples →2
The Baire Principles of Functional AnalysisExamples & counterexamples →2
Distributions Test Functions and DifferentiationExamples & counterexamples →3
Geometric Hahn Banach and Convex SeparationExamples & counterexamples →3
Norming and Separation under Hahn–BanachExamples & counterexamples →3
Dual Spaces Adjoint Operators and AnnihilatorsExamples & counterexamples →4
Locally Convex Spaces and Continuous SeparationExamples & counterexamples →4
Tempered Distributions and the Fourier TransformExamples & counterexamples →4
Weak and Weak Star TopologiesExamples & counterexamples →5
Banach Alaoglu Goldstine and Krein MilmanExamples & counterexamples →6
Reflexivity and Eberlein SmulianExamples & counterexamples →7
Banach Valued Integration and the Radon Nikodym PropertyExamples & counterexamples →8
Schauder Bases Approximation and Banach Space PathologiesExamples & counterexamples →8
Compact Operators and Riesz Schauder TheoryExamples & counterexamples →9
Hilbert Space Geometry and Riesz RepresentationExamples & counterexamples →9
Banach Algebras Spectrum and Holomorphic Functional CalculusExamples & counterexamples →10
Banach-Space Differential Calculus and Banach ManifoldsExamples & counterexamples →10
Orthonormal Bases, Parseval and Fourier SeriesExamples & counterexamples →10
Gelfand Theory and Commutative C Star AlgebrasExamples & counterexamples →11
Square-Integrable Kernels and Hilbert–Schmidt CompactnessExamples & counterexamples →11
Compact Self Adjoint Hilbert Schmidt and Trace Class OperatorsExamples & counterexamples →12
Continuous Functional Calculus for Self Adjoint and Normal OperatorsExamples & counterexamples →12
Spectral Measures and Borel Functional CalculusExamples & counterexamples →13
Unbounded Self Adjoint Operators and Stones TheoremExamples & counterexamples →14

Group Theory

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58 pages · 29 with a worked companion

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Blocks Primitivity and Multiple TransitivityExamples & counterexamples →0
Cayley Graphs, Word Metrics and Quasi-IsometryExamples & counterexamples →0
Crossed Homomorphisms Complements and First CohomologyExamples & counterexamples →0
Decision Problems for Finitely Presented GroupsExamples & counterexamples →0
Extraspecial p-Groups and Central ProductsExamples & counterexamples →0
Group Extensions Complements and Schur ZassenhausExamples & counterexamples →0
Hnn Extensions and Brittons LemmaExamples & counterexamples →0
Modular Representations and Projective CoversExamples & counterexamples →0
Small-Cancellation Disc Diagrams and the Torsion ToolkitExamples & counterexamples →0
Subgroups of Free Groups and Schreier RewritingExamples & counterexamples →0
Asymptotic Cones and the Sublinear Triangle CriterionExamples & counterexamples →1
Boone Machine Simulation and Fixed-Presentation UndecidabilityExamples & counterexamples →1
Brauer Characters and Decomposition MatricesExamples & counterexamples →1
Group Homology Transfer and Low-Degree Exact SequencesExamples & counterexamples →1
Hall–Mal’cev Coordinates and Bass–Guivarc’h GrowthExamples & counterexamples →1
Hyperbolic Spaces and Hyperbolic GroupsExamples & counterexamples →1
Inverse Systems Profinite Groups and CompletionExamples & counterexamples →1
Modular Traces and Brauer-Character IndependenceExamples & counterexamples →1
Second Cohomology and Abelian Kernel ExtensionsExamples & counterexamples →1
Simplicial Trees and Group ActionsExamples & counterexamples →1
Small Cancellation and Dehn AlgorithmsExamples & counterexamples →1
Socles and the Onan Scott LandscapeExamples & counterexamples →1
Geometric Actions Svarc Milnor and GrowthExamples & counterexamples →2
Graphs of Groups and Bass Serre TheoryExamples & counterexamples →2
Pro P Groups and the P Adic IntegersExamples & counterexamples →2
Quantitative Hyperbolic Geometry ToolkitExamples & counterexamples →2
Schur Multipliers and Universal Central ExtensionsExamples & counterexamples →2
The Finite Simple Group Classification LandscapeExamples & counterexamples →2
Amenable Groups and Folner CriteriaExamples & counterexamples →3

Homological Algebra

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36 pages · 18 with a worked companion

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Chain Complexes and HomologyExamples & counterexamples →0
Chain Homotopy and the Homotopy CategoryExamples & counterexamples →1
Spectral SequencesExamples & counterexamples →1
Mapping Cones Cylinders and Chain TrianglesExamples & counterexamples →2
Long Exact Sequences in HomologyExamples & counterexamples →3
Double Complexes Exact Couples and ConvergenceExamples & counterexamples →4
Projective and Injective ResolutionsExamples & counterexamples →4
Triangulated CategoriesExamples & counterexamples →4
Derived FunctorsExamples & counterexamples →5
Delta Functors and UniversalityExamples & counterexamples →6
Ext and Balanced ResolutionsExamples & counterexamples →7
Yoneda Extensions and Homological DimensionExamples & counterexamples →8
Group Cohomology as a Derived FunctorExamples & counterexamples →9
Tor Flatness and Global DimensionExamples & counterexamples →9
Derived CategoriesExamples & counterexamples →10
Universal Coefficients and Kunneth TheoremsExamples & counterexamples →10
Grothendieck Spectral Sequences and ComputationsExamples & counterexamples →11
Kunneth Exactness and Splittings over Principal Ideal DomainsExamples & counterexamples →11

Lie Theory

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18 pages · 9 with a worked companion

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Finite Weyl Invariants, Bruhat Order, and Kostant HarmonicsExamples & counterexamples →0
Harish Chandra Isomorphism Casimir and Central CharactersExamples & counterexamples →0
Kac Moody Algebras from Generalized Cartan MatricesExamples & counterexamples →1
Verma Modules and Shapovalov FormsExamples & counterexamples →1
Affine Lie Algebras and Loop Central ExtensionsExamples & counterexamples →2
Homomorphisms Between Verma Modules and LinkageExamples & counterexamples →2
Integrable Highest Weight Kac Moody ModulesExamples & counterexamples →2
Category O Finiteness Duality and BlocksExamples & counterexamples →3
The Weyl Kac Character FormulaExamples & counterexamples →3

Linear Algebra

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44 pages · 22 with a worked companion

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Vector Spaces, Linear Subspaces, Span and Direct SumsExamples & counterexamples →0
Linear Independence, Bases and DimensionExamples & counterexamples →1
Linear Transformations, Rank-Nullity and Quotient SpacesExamples & counterexamples →2
Matrices, the Matrix of a Linear Map, and Change of BasisExamples & counterexamples →3
Gaussian Elimination, Elementary Matrices and Reduced Row Echelon FormExamples & counterexamples →4
Determinants of Matrices over a Commutative RingExamples & counterexamples →5
Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of InertiaExamples & counterexamples →6
The Determinant of a Linear Operator, Cofactors and Cramer's RuleExamples & counterexamples →6
Eigenvalues, Eigenvectors and the Characteristic PolynomialExamples & counterexamples →7
Inner Product Spaces, Gram-Schmidt, Projections and AdjointsExamples & counterexamples →7
Diagonalisation and the Minimal PolynomialExamples & counterexamples →8
Complexification, Realification and Real StructuresExamples & counterexamples →9
Triangularisation, Generalised Eigenspaces and Jordan Canonical FormExamples & counterexamples →9
Exterior Powers, Orientation and Hodge DualityExamples & counterexamples →10
The Spectral Theorem, Positive Operators and Singular Value DecompositionExamples & counterexamples →10
Matrix Norms, Condition Numbers and Numerical StabilityExamples & counterexamples →11
Direct Matrix Factorisations: LU, Cholesky and QRExamples & counterexamples →12
Matrix Differentiation and First-order Spectral PerturbationExamples & counterexamples →12
Eigenvalue Iterations and the QR AlgorithmExamples & counterexamples →13
The Moore--Penrose Pseudoinverse and Regularised Least SquaresExamples & counterexamples →13
Krylov Subspaces, Arnoldi and GMRESExamples & counterexamples →14
Conjugate Gradients, MINRES and PreconditioningExamples & counterexamples →15

Measure Theory

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54 pages · 27 with a worked companion

PageCompanionDependency level
Sigma Algebras and Borel SetsExamples & counterexamples →0
Measures and Their Basic PropertiesExamples & counterexamples →1
Outer Measure and the Caratheodory Extension TheoremExamples & counterexamples →2
Lebesgue Measure on Euclidean SpaceExamples & counterexamples →3
Lebesgue-Stieltjes Measures and Distribution FunctionsExamples & counterexamples →4
Non Measurable Sets and the Cost of ChoiceExamples & counterexamples →4
The Lebesgue Integral and the Convergence TheoremsExamples & counterexamples →4
Measurable Functions and Simple ApproximationExamples & counterexamples →5
Hausdorff Measure and Hausdorff DimensionExamples & counterexamples →6
Modes of Convergence Egorov and LusinExamples & counterexamples →6
Product Measures and the Fubini Tonelli TheoremsExamples & counterexamples →6
Signed and Complex Measures Hahn and JordanExamples & counterexamples →6
The Lebesgue and Riemann Integrals ComparedExamples & counterexamples →6
The Lᵖ Spaces Holder Minkowski and Riesz FischerExamples & counterexamples →7
The Radon Nikodym Theorem and Lebesgue DecompositionExamples & counterexamples →7
Density Separability and Convolution in LᵖExamples & counterexamples →8
The Duality of Lᵖ and L^qExamples & counterexamples →8
Complex Lp Spaces and Test-Function ConventionsExamples & counterexamples →9
Radon Measures and the Riesz Markov Kakutani TheoremExamples & counterexamples →9
The Maximal Function and Lebesgue DifferentiationExamples & counterexamples →9
Complex Riesz–Thorin Endpoint InterpolationExamples & counterexamples →10
Differentiation of Monotone Functions and the Vitali Covering TheoremExamples & counterexamples →10
Measure-Preserving Systems and Mixing CriteriaExamples & counterexamples →10
Absolute Continuity and the Sharp Fundamental Theorem of CalculusExamples & counterexamples →11
Measure Preserving Transformations and Poincare RecurrenceExamples & counterexamples →11
Weak Mixing and the Chacon TransformationExamples & counterexamples →11
The Ergodic Theorems of von Neumann and BirkhoffExamples & counterexamples →12

Number Theory

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48 pages · 24 with a worked companion

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Divisibility, Greatest Common Divisors and Bézout's IdentityExamples & counterexamples →0
Primes, Euclid's Lemma and the Fundamental Theorem of ArithmeticExamples & counterexamples →1
Regular Continued Fractions and Diophantine ApproximationExamples & counterexamples →1
Congruences, the Integers Modulo n and the Chinese Remainder TheoremExamples & counterexamples →2
Pell Equations and Generalized Pell OrbitsExamples & counterexamples →2
Absolute Values Completions and P Adic NumbersExamples & counterexamples →3
Arithmetic Functions and Dirichlet ConvolutionExamples & counterexamples →3
Number Fields Rings of Integers and DiscriminantsExamples & counterexamples →3
Positive Definite Binary Quadratic Forms and ReductionExamples & counterexamples →3
Primitive Roots and Unit Groups Modulo NExamples & counterexamples →3
Dirichlet Series and Euler ProductsExamples & counterexamples →4
Prime Ideal Decomposition Ramification and the DifferentExamples & counterexamples →4
Quadratic Residues and the Legendre SymbolExamples & counterexamples →4
Decomposition Inertia and FrobeniusExamples & counterexamples →5
Quadratic Reciprocity and the Jacobi SymbolExamples & counterexamples →5
Sums of Two SquaresExamples & counterexamples →5
Average Orders Divisor Sums and Representation CountsExamples & counterexamples →6
Lagrange Four Square TheoremExamples & counterexamples →6
Chebyshev Bounds and Mertens TheoremsExamples & counterexamples →7
Dirichlet Characters L Functions and Primes in ProgressionsExamples & counterexamples →8
Perron Inversion and the Explicit FormulaExamples & counterexamples →8
Classical Zero Free Region and the Prime Number TheoremExamples & counterexamples →9
Hilbert Symbols and the Quadratic Local Global PrincipleExamples & counterexamples →9
Primitive Dirichlet L Functions and Functional EquationsExamples & counterexamples →9

Partial Differential Equations

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12 pages · 6 with a worked companion

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Euclidean Surface Measure, Divergence, and Green IdentitiesExamples & counterexamples →0
Harmonic Functions and Mean Values in RnExamples & counterexamples →0
Partial Differential Equations and CharacteristicsExamples & counterexamples →0
Analytic Majorants and the Cauchy–Kovalevskaya TheoremExamples & counterexamples →1
Maximum Principles Harnack and Liouville in RnExamples & counterexamples →1
Quasilinear Characteristics and Cauchy KovalevskayaExamples & counterexamples →1

Probability

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42 pages · 21 with a worked companion

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Probability Spaces Random Variables and ExpectationExamples & counterexamples →0
Conditional ExpectationExamples & counterexamples →1
Independence Borel Cantelli and Zero One LawsExamples & counterexamples →1
Discrete Time MartingalesExamples & counterexamples →2
Infinite Product Measures and Kolmogorov ExtensionExamples & counterexamples →2
Modes of Convergence for Random VariablesExamples & counterexamples →2
Standard-Borel Real Codings and Determining ClassesExamples & counterexamples →3
Weak Laws and Series of Independent Random VariablesExamples & counterexamples →3
Conditional Distributions and Regular Conditional ProbabilityExamples & counterexamples →4
Strong Laws of Large NumbersExamples & counterexamples →4
Weak Convergence Tightness and RepresentationExamples & counterexamples →5
Characteristic Functions Inversion and ContinuityExamples & counterexamples →6
Central Limit TheoremsExamples & counterexamples →7
Brownian Motion Construction and ContinuityExamples & counterexamples →8
Martingale Inequalities and ConvergenceExamples & counterexamples →8
Stopping Times and Optional StoppingExamples & counterexamples →9
Brownian Motion, Markov Properties and Hitting TimesExamples & counterexamples →10
Markov Kernels and Markov ChainsExamples & counterexamples →10
Brownian Path PropertiesExamples & counterexamples →11
The Ito Integral with Respect to Brownian MotionExamples & counterexamples →12
Itos Formula and Brownian MartingalesExamples & counterexamples →13

Real Analysis

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109 pages · 51 with a worked companion

PageCompanionDependency level
Construction of the Real Numbers via Cauchy Sequences0
Construction of the Real Numbers via Dedekind Cuts0
Foundations of the Real Numbers for Analysis1
Suprema and InfimaExamples & counterexamples →2
Countability and Uncountability3
Roots, Rational Powers, and Classical Inequalities3
Sequences and Limits4
Monotone Sequences, Bolzano-Weierstrass, and Cauchy CompletenessExamples & counterexamples →5
limsup, liminf, and Subsequential LimitsExamples & counterexamples →6
The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete6
Topology of ℝExamples & counterexamples →6
Equivalent Forms of CompletenessExamples & counterexamples →7
Limits of Real FunctionsExamples & counterexamples →7
Series: Convergence and the Nonnegative TestsExamples & counterexamples →7
Absolute and Conditional Convergence; Rearrangement; ProductsExamples & counterexamples →8
Continuity, IVT, EVT, and Uniform ContinuityExamples & counterexamples →8
The Cantor Set, Baire Category, and Measure Zero in ℝExamples & counterexamples →8
Monotone Functions, Discontinuities, and Continuity SetsExamples & counterexamples →9
The Derivative and the Mean Value TheoremsExamples & counterexamples →10
The Riemann Integral: Definition and IntegrabilityExamples & counterexamples →10
Darboux, L'Hôpital, and Taylor's TheoremExamples & counterexamples →11
Properties of the Integral and the Working FTCExamples & counterexamples →11
The Gauge Integral and Cousin's LemmaExamples & counterexamples →11
Bounded Variation and the Riemann–Stieltjes IntegralExamples & counterexamples →12
ConvexityExamples & counterexamples →12
Improper IntegralsExamples & counterexamples →12
Rⁿ as a Normed Space; Vector-Valued FunctionsExamples & counterexamples →12
Sequences and Series of Functions; Uniform ConvergenceExamples & counterexamples →12
Approximation and Compactness in C(K)Examples & counterexamples →13
Arc Length and Rectifiable CurvesExamples & counterexamples →13
Power Series and Real-Analytic FunctionsExamples & counterexamples →13
The Fundamental Theorems of CalculusExamples & counterexamples →13
The Riemann Integral in Rᵐ and Jordan ContentExamples & counterexamples →13
The Total Derivative in ℝᵐ → ℝⁿExamples & counterexamples →13
Sine, Cosine, and the Definition of PiExamples & counterexamples →14
The Exponential FunctionExamples & counterexamples →14
The Inverse and Implicit Function TheoremsExamples & counterexamples →14
Fubini and Change of VariablesExamples & counterexamples →15
Fundamental Trigonometric IdentitiesExamples & counterexamples →15
Mixed Partials, Taylor Formulae, and ExtremaExamples & counterexamples →15
pi: the Equivalent CharacterizationsExamples & counterexamples →15
Picard-Lindelöf and First-Order Ordinary Differential EquationsExamples & counterexamples →15
The Logarithm and General PowersExamples & counterexamples →15
Areas of Elementary Plane FiguresExamples & counterexamples →16
Convex and Semicontinuous Functions on RⁿExamples & counterexamples →16
Further Trigonometric Identities and Inverse FunctionsExamples & counterexamples →16
Improper and Parameter-Dependent Multiple IntegralsExamples & counterexamples →16
Line Integrals and the Gradient TheoremExamples & counterexamples →16
The Complex Exponential and Euler's FormulaExamples & counterexamples →16
The Integral Logarithm and the Equivalence of Its CharacterisationsExamples & counterexamples →16
The Inverse Function Theorem CompletedExamples & counterexamples →16
Trigonometric and Oscillatory Examples in Several VariablesExamples & counterexamples →16
Constant Rank, Submersions, Immersions and Regular Level SetsExamples & counterexamples →17
Trigonometric and Oscillatory Examples in One VariableExamples & counterexamples →17
Volumes of Elementary Solids and Solids of RevolutionExamples & counterexamples →17
Regular Surfaces and Surface IntegralsExamples & counterexamples →18
The Real Gamma and Beta FunctionsExamples & counterexamples →18
The Divergence Theorem and Classical StokesExamples & counterexamples →19

Recorded, Not Proved Here

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3 pages

PageCompanionDependency level
Measure and Integration: Recorded, Not Proved Here0
Set Theory Beyond Choice: Recorded, Not Proved Here0
Open Problems and the Research Frontier1

Representation Theory of Groups

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20 pages · 10 with a worked companion

PageCompanionDependency level
Artin Induction and Rational CharactersExamples & counterexamples →0
Blocks Defect Groups and the Brauer HomomorphismExamples & counterexamples →0
Clifford Theory over Normal SubgroupsExamples & counterexamples →0
Galois Orbits and Descent of Simple Finite-Group ModulesExamples & counterexamples →0
Haar Measure Existence and UniquenessExamples & counterexamples →0
Brauer Induction and Elementary SubgroupsExamples & counterexamples →1
Vertices Sources and the Green CorrespondenceExamples & counterexamples →1
Brauers First Main TheoremExamples & counterexamples →2
Schur Indices and Fields of DefinitionExamples & counterexamples →2
Brauers Second Main TheoremExamples & counterexamples →3

Scheme Theory

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10 pages · 5 with a worked companion

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Presheaves Sheaves Stalks and SheafificationExamples & counterexamples →0
Sheaf Operations Exactness Ringed Spaces and Module PullbackExamples & counterexamples →1
Affine Schemes and the Structure SheafExamples & counterexamples →2
Schemes Subschemes and Morphisms Locally of Finite TypeExamples & counterexamples →3
Fibre Products Base Change and Scheme Theoretic FibresExamples & counterexamples →4

Set Theory

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78 pages · 38 with a worked companion

PageCompanionDependency level
The ZFC Axioms and the Basic Set ConstructionsExamples & counterexamples →0
Relations, Functions, and QuotientsExamples & counterexamples →1
Construction of the Natural Numbers2
Dependent Choice and the Complete-Metric Baire TheoremExamples & counterexamples →3
Formal Set-Theoretic Syntax, Structures, and SatisfactionExamples & counterexamples →3
Order, Zorn's Lemma, and the Axiom of ChoiceExamples & counterexamples →3
Filters and UltrafiltersExamples & counterexamples →4
Ordinals, Cardinals, and Transfinite Recursion4
Ordinal Arithmetic and the First Uncountable OrdinalExamples & counterexamples →5
Well-Founded Relations, Rank, and the Cumulative HierarchyExamples & counterexamples →5
Cardinal Arithmetic, Cofinality and the AlephsExamples & counterexamples →6
Borel and Analytic Sets, Perfect Sets, and DeterminacyExamples & counterexamples →7
Club, Stationary Sets, and Pressing DownExamples & counterexamples →7
Deduction, Soundness, Completeness, and CompactnessExamples & counterexamples →7
Weak Choice Principles and Sierpiński's TheoremExamples & counterexamples →7
Arithmetization, Incompleteness, and Relative ConsistencyExamples & counterexamples →8
Boolean Algebras, Stone Duality, and the Prime Ideal TheoremExamples & counterexamples →8
Reflection, Absoluteness, and Elementary SubmodelsExamples & counterexamples →8
Set-Theoretic Trees, Delta Systems, and DiamondExamples & counterexamples →8
Forcing Orders, Names, and Generic ExtensionsExamples & counterexamples →9
PCF Scales and ZFC Dowker SpacesExamples & counterexamples →9
The Constructible Hierarchy and Inner ModelsExamples & counterexamples →9
Condensation, GCH, and Diamond in LExamples & counterexamples →10
Large Cardinals, Measures, and Elementary EmbeddingsExamples & counterexamples →10
The Forcing Theorem and Formal Consistency TransferExamples & counterexamples →10
Permutation Models and Transfer to ZFExamples & counterexamples →11
Preservation, Cohen Forcing, and the ContinuumExamples & counterexamples →11
Finite-Support Iterations and Martin's AxiomExamples & counterexamples →12
Solovay's Model and Regularity of All Sets of RealsExamples & counterexamples →12
Symmetric Extensions and Basic Choice-Failure ModelsExamples & counterexamples →12
Boolean Prime Ideal Theorem in the Basic Cohen ModelExamples & counterexamples →13
Choice Strength in Baire, Urysohn, Stone, and TychonoffExamples & counterexamples →13
Normal Moore Spaces, PMEA, and Consistency StrengthExamples & counterexamples →13
Prikry Forcing and Gitik's Singular-Cardinal ModelExamples & counterexamples →13
Suslin Trees, Lines, Algebras, and IndependenceExamples & counterexamples →13
Symmetric Collapse and Ultrafilter-Free ModelsExamples & counterexamples →13
Halpern–Läuchli and BPI Without ChoiceExamples & counterexamples →14
Proper Forcing, Countable-Support Iterations, and PFAExamples & counterexamples →14
Shelah's Baire-Property Model and Inner-Model Lower BoundsExamples & counterexamples →14
Minimal Walks, Oscillation, and L- and S-SpacesExamples & counterexamples →15

Topology

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60 pages · 30 with a worked companion

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Metric SpacesExamples & counterexamples →0
Completeness, Completion, and Uniform ContinuityExamples & counterexamples →1
Topological Spaces and ContinuityExamples & counterexamples →1
Compactness in Metric SpacesExamples & counterexamples →2
Subspaces, Products, and QuotientsExamples & counterexamples →2
ConnectednessExamples & counterexamples →3
CompactnessExamples & counterexamples →4
Convergence: Nets and FiltersExamples & counterexamples →5
Countability Axioms and Cardinal FunctionsExamples & counterexamples →5
Function Space Topologies and the Exponential LawExamples & counterexamples →5
Separation Axioms: the HierarchyExamples & counterexamples →5
The Topology of Euclidean SpaceExamples & counterexamples →5
Hausdorff via the DiagonalExamples & counterexamples →6
Homotopy and Homotopy EquivalenceExamples & counterexamples →6
Stone–Weierstrass in GeneralExamples & counterexamples →6
The Ascoli–Arzelà TheoremExamples & counterexamples →6
Hereditary and Productive Behaviour of the Separation AxiomsExamples & counterexamples →7
The Fundamental GroupExamples & counterexamples →7
Uniform Spaces: the Three DefinitionsExamples & counterexamples →7
Urysohn's Lemma and the Tietze Extension TheoremExamples & counterexamples →7
Covering Spaces and LiftingExamples & counterexamples →8
Partitions of Unity and ParacompactnessExamples & counterexamples →8
The Tychonoff Embedding and the Stone–Čech CompactificationExamples & counterexamples →8
Complete Metrizability, Čech-Completeness, and Baire CategoryExamples & counterexamples →9
Metrization: Urysohn, Nagata–Smirnov, Bing, SmirnovExamples & counterexamples →9
The Fundamental Group of the CircleExamples & counterexamples →9
Uniform Completeness, Completion, and the Samuel CompactificationExamples & counterexamples →9
Classification of Covering SpacesExamples & counterexamples →10
The Seifert–van Kampen TheoremExamples & counterexamples →10
Applications of the Fundamental GroupExamples & counterexamples →11