Editing Equivalence class (section)

==Quotient space in topology==

In [[topology]], a [[Quotient space (topology)|quotient space]] is a [[topological space]] formed on the set of equivalence classes of an equivalence relation on a topological space, using the original space's topology to create the topology on the set of equivalence classes.

In [[abstract algebra]], [[congruence relation]]s on the underlying set of an algebra allow the algebra to induce an algebra on the equivalence classes of the relation, called a [[Quotient (universal algebra)|quotient algebra]]. In [[linear algebra]], a [[Quotient space (linear algebra)|quotient space]] is a vector space formed by taking a [[quotient group]], where the quotient homomorphism is a [[linear map]]. By extension, in abstract algebra, the term quotient space may be used for [[quotient module]]s, [[quotient ring]]s, [[quotient group]]s, or any quotient algebra. However, the use of the term for the more general cases can as often be by analogy with the orbits of a group action.

The orbits of a [[Group action (mathematics)|group action]] on a set may be called the quotient space of the action on the set, particularly when the orbits of the group action are the right [[coset]]s of a subgroup of a group, which arise from the action of the subgroup on the group by left translations, or respectively the left cosets as orbits under right translation.

A normal subgroup of a topological group, acting on the group by translation action, is a quotient space in the senses of topology, abstract algebra, and group actions simultaneously.

Although the term can be used for any equivalence relation's set of equivalence classes, possibly with further structure, the intent of using the term is generally to compare that type of equivalence relation on a set <math>X,</math> either to an equivalence relation that induces some structure on the set of equivalence classes from a structure of the same kind on <math>X,</math> or to the orbits of a group action. Both the sense of a structure preserved by an equivalence relation, and the study of [[Invariant (mathematics)|invariants]] under group actions, lead to the definition of [[#Invariants|invariants]] of equivalence relations given above.