Difference between classification and characterization What is the difference between classification and characterization in reference to mathematical objects ?
Some examples will be appreciated.
 A: A type of mathematical object can be classified into different categories or types; each example of the object lies in exactly one category (or is of exactly one type). This is what we call a classification. Given a type of mathematical object, it can also be classified in different ways. Here are some examples:


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*Classification of finite simple groups.

*Classification of semisimple Lie algebras over an algebraically closed field.

*Classification of one-dimensional manifolds.

*Classification of closed surfaces (connected, compact, two-dimensional manifolds without boundary).

*Classification of simply connected Riemann surfaces.


A mathematical object (or collection of mathematical objects) can be characterised by a collection of properties; that is, it is determined by those properties alone. This is what we call a characterisation. In particular, a type of mathematical object is characterised by its definition, but we are often interested in other characterisations. Here are some examples:


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*The prime numbers are characterised by their definition: numbers greater than one which have only two factors. However, they are also characterised by the following property: numbers $p$ greater than one such that if $p \mid ab$, then $p \mid a$ or $p \mid b$.

*The invertible matrices are characterised by the properties of being square and having non-zero determinant.

*Infinite sets are characterised by the property of having a proper subset with which it is equinumerous (i.e. there is a bijection between the subset and the set itself).

*Harmonic functions are characterised by the mean value property (assuming local integrability).


There are many more examples for both lists. If I think of any more instructive ones I'll add them.
Added Later: You might find the Wikipedia pages on classification and characterisation useful as well.
