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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>classification theorem</b> answers the <a href="Classification" title="Classification">classification</a> problem: "What are the objects of a given type, up to some <a href="Equivalence_relation" title="Equivalence relation">equivalence</a>?". It gives a non-redundant <a href="Enumeration" title="Enumeration">enumeration</a>: each object is equivalent to exactly one class.

</p><p>A few issues related to classification are the following.

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<ul><li>The equivalence problem is "given two objects, determine if they are equivalent".</li>

<li>A <a href="Complete_set_of_invariants" title="Complete set of invariants">complete set of invariants</a>, together with which invariants are realizable, solves the classification problem, and is often a step in solving it. (A combination of invariant values is realizable if there in fact exists an object whose invariants take on the specified set of values)</li>

<li>A <span class="clarify-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">computable complete set of invariants</span> (together with which invariants are realizable) solves both the classification problem and the equivalence problem.</li>

<li>A <a href="Canonical_form" title="Canonical form">canonical form</a> solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class.</li></ul>

<p>There exist many <b>classification theorems</b> in <a href="Mathematics" title="Mathematics">mathematics</a>, as described below.

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<div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2></div>

<ul><li><a href="Euclidean_plane_isometry#Classification_of_Euclidean_plane_isometries" title="Euclidean plane isometry">Classification of Euclidean plane isometries</a></li>

<li><a href="Platonic_solid#Classification" title="Platonic solid">Classification of Platonic solids</a></li>

<li>Classification theorems of surfaces

<ul><li><a href="Classification_of_two-dimensional_closed_manifolds" class="mw-redirect" title="Classification of two-dimensional closed manifolds">Classification of two-dimensional closed manifolds</a> – Two-dimensional manifold<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Enriques%E2%80%93Kodaira_classification" title="Enriques–Kodaira classification">Enriques–Kodaira classification</a> – Mathematical classification of surfaces of <a href="Algebraic_surfaces" class="mw-redirect" title="Algebraic surfaces">algebraic surfaces</a> (complex dimension two, real dimension four)</li>

<li><a href="Nielsen%E2%80%93Thurston_classification" title="Nielsen–Thurston classification">Nielsen–Thurston classification</a> – Characterizes homeomorphisms of a compact orientable surface which characterizes homeomorphisms of a compact surface</li></ul></li>

<li>Thurston's eight model geometries, and the <a href="Geometrization_conjecture" title="Geometrization conjecture">geometrization conjecture</a> – Three dimensional analogue of uniformization conjecture</li>

<li><a href="Holonomy#The_Berger_classification" title="Holonomy">Berger classification</a> – Concept in differential geometry</li>

<li><a href="Symmetric_space#Classification_result" title="Symmetric space">Classification of Riemannian symmetric spaces</a> – (pseudo-)Riemannian manifold whose geodesics are reversible</li>

<li><a href="Lens_space#Classification_of_3-dimensional_lens_spaces" title="Lens space">Classification of 3-dimensional lens spaces</a> – Class of topological space</li>

<li><a href="Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a> – Basic question in geometry and topology</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Algebra">Algebra</h2></div>

<ul><li><a href="Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a> – Theorem classifying finite simple groups

<ul><li><a href="Abelian_group#Classification" title="Abelian group">Classification of Abelian groups</a> – Commutative group (mathematics)</li>

<li><a href="Finitely_generated_abelian_group#Classification" title="Finitely generated abelian group">Classification of Finitely generated abelian group</a> – Commutative group where every element is the sum of elements from one finite subset</li>

<li><a href="Multiple_transitivity" class="mw-redirect" title="Multiple transitivity">Classification of Rank 3 permutation group</a> – Five sporadic simple groups<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li>

<li><a href="Rank_3_permutation_group#Classification" title="Rank 3 permutation group">Classification of 2-transitive permutation groups</a></li></ul></li>

<li><a href="Artin%E2%80%93Wedderburn_theorem" class="mw-redirect" title="Artin–Wedderburn theorem">Artin–Wedderburn theorem</a> – Classification of semi-simple rings and algebras<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span> — a classification theorem for semisimple rings</li>

<li><a href="Classification_of_Clifford_algebras" title="Classification of Clifford algebras">Classification of Clifford algebras</a></li>

<li><a href="Classification_of_low-dimensional_real_Lie_algebras" title="Classification of low-dimensional real Lie algebras">Classification of low-dimensional real Lie algebras</a></li>

<li>Classification of Simple Lie algebras and groups

<ul><li><a href="Semisimple_Lie_algebra#Classification" title="Semisimple Lie algebra">Classification of simple complex Lie algebras</a> – Direct sum of simple Lie algebras</li>

<li><a href="Satake_diagram" title="Satake diagram">Classification of simple real Lie algebras</a> – Term in mathematics</li>

<li><a href="Simple_Lie_group#Full_classification" title="Simple Lie group">Classification of centerless simple Lie groups</a> – Connected non-abelian Lie group lacking nontrivial connected normal subgroups</li>

<li><a href="List_of_simple_Lie_groups" class="mw-redirect" title="List of simple Lie groups">Classification of simple Lie groups</a> – Connected non-abelian Lie group lacking nontrivial connected normal subgroups<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul></li>

<li><a href="Bianchi_classification" title="Bianchi classification">Bianchi classification</a> – Lie algebra classification</li>

<li><a href="ADE_classification" title="ADE classification">ADE classification</a> – Mathematical classification</li>

<li><a href="Langlands_classification" title="Langlands classification">Langlands classification</a> – Mathematical theory</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Linear_algebra">Linear algebra</h2></div>

<ul><li><a href="Finite-dimensional_vector_space" class="mw-redirect" title="Finite-dimensional vector space">Finite-dimensional vector space</a> – Number of vectors in any basis of the vector space<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span>s (by dimension)</li>

<li><a href="Rank%E2%80%93nullity_theorem" title="Rank–nullity theorem">Rank–nullity theorem</a> – In linear algebra, relation between 3 dimensions (by rank and nullity)</li>

<li><a href="Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain" title="Structure theorem for finitely generated modules over a principal ideal domain">Structure theorem for finitely generated modules over a principal ideal domain</a> – Statement in abstract algebra</li>

<li><a href="Jordan_normal_form" title="Jordan normal form">Jordan normal form</a> – Form of a matrix indicating its eigenvalues and their algebraic multiplicities</li>

<li><a href="Frobenius_normal_form" title="Frobenius normal form">Frobenius normal form</a> – Canonical form of matrices over a field (rational canonical form)</li>

<li><a href="Sylvester's_law_of_inertia" title="Sylvester's law of inertia">Sylvester's law of inertia</a> – Theorem of matrix algebra of invariance properties under basis transformations</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Analysis">Analysis</h2></div>

<ul><li><a href="Classification_of_discontinuities" title="Classification of discontinuities">Classification of discontinuities</a> – Mathematical analysis of discontinuous points</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Dynamical_systems">Dynamical systems</h2></div>

<ul><li><a href="Classification_of_Fatou_components" title="Classification of Fatou components">Classification of Fatou components</a></li>

<li><a href="Ratner's_theorems#Short_description" title="Ratner's theorems">Ratner classification theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_physics">Mathematical physics</h2></div>

<ul><li><a href="Classification_of_electromagnetic_fields" title="Classification of electromagnetic fields">Classification of electromagnetic fields</a></li>

<li><a href="Petrov_classification" title="Petrov classification">Petrov classification</a> – Classification used in differential geometry and general relativity</li>

<li><a href="Segre_classification" title="Segre classification">Segre classification</a> – Algebraic classification of rank two symmetric tensors</li>

<li><a href="Wigner's_classification" title="Wigner's classification">Wigner's classification</a> – Classification of irreducible representations of the Poincaré group</li></ul>

<ul><li><a href="Representation_theorem" title="Representation theorem">Representation theorem</a> – Proof that every structure with certain properties is isomorphic to another structure</li>

<li><a href="Comparison_theorem" title="Comparison theorem">Comparison theorem</a></li>

<li><a href="List_of_manifolds" title="List of manifolds">List of manifolds</a></li>

<li><a href="List_of_theorems" title="List of theorems">List of theorems</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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