# Classification theorem

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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 related issues 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.

## Contents

## GeometryEdit

- 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

## AlgebraEdit

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