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 combination of invariant values is realizable if there in fact exists an object whose invariants take on the specified set of values)

• A (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 Platonic solids

Classification theorems of surfaces

• of algebraic surfaces (complex dimension two, real dimension four)

• which characterizes homeomorphisms of a compact surface

• Thurston's eight model geometries, and the

Algebra

• a classification theorem for semisimple rings

Classification of Simple Lie algebras and groups

Linear algebra

• s (by dimension)

• (by rank and nullity)

• (rational canonical form)

Analysis

Dynamical systems

• Ratner classification theorem

Mathematical physics

See also

• List of theorems

References