Editing Isomorphism (section)

=== Examples ===
Examples of isomorphism classes are plentiful in mathematics.
* Two sets are isomorphic if there is a [[bijection]] between them. The isomorphism class of a finite set can be identified with the non-negative integer representing the number of elements it contains.
* The isomorphism class of a [[finite-dimensional vector space]] can be identified with the non-negative integer representing its dimension.
* The [[classification of finite simple groups]] enumerates the isomorphism classes of all [[finite simple groups]].
* The [[Surface (topology)#Classification of closed surfaces|classification of closed surfaces]] enumerates the isomorphism classes of all connected [[closed surface]]s.
* [[ordinal number|Ordinals]] are essentially defined as isomorphism classes of well-ordered sets (though there are technical issues involved).
* There are three isomorphism classes of the planar [[subalgebra]]s of M(2,'''R'''), the 2 x 2 real matrices.

However, there are circumstances in which the isomorphism class of an object conceals vital  information about it.

* Given a [[mathematical structure]], it is common that two [[substructure (mathematics)|substructure]]s belong to the same isomorphism class. However, the way they are  included in the whole structure can not be studied if they are identified. For example, in a finite-dimensional vector space, all [[Linear subspace|subspaces]] of the same dimension are isomorphic, but must be distinguished to consider their intersection, sum, etc.
* In [[homotopy theory]], the [[fundamental group]] of a [[topological space|space]] <math>X</math> at a point <math>p</math>, though technically denoted <math>\pi_1(X,p)</math> to emphasize the dependence on the base point, is often written lazily as simply <math>\pi_1(X)</math> if <math>X</math> is [[connected space#Path connectedness|path connected]]. The reason for this is that the existence of a path between two points allows one to identify [[loop (topology)|loops]] at one with loops at the other; however, unless <math>\pi_1(X,p)</math> is [[abelian group|abelian]] this isomorphism is non-unique. Furthermore, the classification of [[covering space]]s makes strict reference to particular [[subgroup]]s of <math>\pi_1(X,p)</math>, specifically distinguishing between isomorphic but [[conjugacy class|conjugate]] subgroups, and therefore amalgamating the elements of an isomorphism class into a single featureless object seriously decreases the level of detail provided by the theory.