Classification theorem: Difference between revisions

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{{short description|Describes the objects of a given type, up to some equivalence}}

{{Unreferenced|date=December 2009}}

In [[mathematics]], a '''classification theorem''' answers the [[classification]] problem: "What are the objects of a given type, up to some [[Equivalence relation|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 {{clarify span|realizable,|reason=This notion should be introduced.|date=October 2020}} 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 {{clarify span|computable complete set of invariants|reason=Shouldn't this be "finite set of computable invariants"? Computability (whatever this is supposed to mean on a set of functions) is of no help if infinitely many functions must be evaluated or if an uncomputable function must be evaluated.|date=October 2020}} (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.

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==Geometry==

* {{annotated link|Euclidean plane isometry#Classification of Euclidean plane isometries|Classification of Euclidean plane isometries}}

* [[Platonic solid#Classification|Classification of Platonic solids]]

* Classification theorems of surfaces

** {{annotated link|Classification of two-dimensional closed manifolds}}

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* {{annotated link|Comparison theorem}}

* {{annotated link|List of manifolds}}

* [[List of theorems]]

==References==

{{reflist}}

{{DEFAULTSORT:Classification Theorem}}