In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if and only if a and b are equivalent. Formally, given a set S and an equivalence relation ~ on S, the equivalence class of an element a in S is the set of elements which are equivalent to a. It may be proven from the defining properties of "equivalence relations" that the equivalence classes form a partition of X. This partition – the set of equivalence classes – is sometimes called the quotient set or the quotient space of X by ~ and is denoted by X / ~. When the set S has some structure (such as a group operation or a topology) and the equivalence relation ~ is defined in a manner suitably compatible with this structure, then the quotient set often inherits a similar structure from its parent set. Examples include quotient spaces in linear algebra, quotient spaces in topology, quotient groups, homogeneous spaces, quotient rings, quotient monoids, and quotient categories.

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