Classification theorem

In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.

A few issues related to classification are the following.

The equivalence problem is "given two objects, determine if they are equivalent".
A complete set of invariants, together with which invariants are realizable, solves the classification problem, and is often a step in solving it.
A computable complete set of invariants (together with which invariants are realizable) solves both the classification problem and the equivalence problem.
A canonical form solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class.

There exist many classification theorems in mathematics, as described below.

Geometry

Classification of Euclidean plane isometries
Classification theorem of surfaces
- Classification of two-dimensional closed manifolds
- Enriques–Kodaira classification of algebraic surfaces (complex dimension two, real dimension four)
- Nielsen–Thurston classification which characterizes homeomorphisms of a compact surface
Thurston's eight model geometries, and the geometrization conjecture

Algebra

Classification of finite simple groups
Artin–Wedderburn theorem — a classification theorem for semisimple rings

Linear algebra

Finite-dimensional vector spaces (by dimension)
rank–nullity theorem (by rank and nullity)
Structure theorem for finitely generated modules over a principal ideal domain
Jordan normal form
Sylvester's law of inertia

Complex analysis

Classification of Fatou components

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