Definition:Quotient

Algebra

 * The quotient of $$a$$ on division by $$b$$ is the unique number $$q$$ such that $$a = q b + r, 0 \le r < \left|{b}\right|$$ (see the Division Theorem).

Set theory

 * Quotient Set: The set $$S / \mathcal R$$ of $\mathcal R$-classes of an equivalence relation $$\mathcal R$$ of a set $$S$$.


 * Quotient Mapping: The mapping $$q_{\mathcal R}: S \to S / \mathcal R$$ defined as $$q_{\mathcal R} \left({s}\right) = \left[\!\left[{s}\right]\!\right]_{\mathcal R}$$.

Abstract Algebra
The concepts here, although presented in different forms, are all related.


 * Quotient Structure: If $$\mathcal R$$ is a congruence for $$\circ$$ on an algebraic structure $$\left({S, \circ}\right)$$, and $$\circ_{\mathcal R}$$ is the operation induced on $S / \mathcal R$ by $\circ$, then $$\left({S / \mathcal R, \circ_{\mathcal R}}\right)$$ is the quotient structure defined by $\mathcal R$.


 * Quotient Group: The coset space $$G / N$$, where $$N$$ is a normal subgroup of a group $$G$$, and the group product is defined as $$\left({a N}\right) \left({b N}\right) = \left({a b}\right) N$$ is called the quotient group of $$G$$ by $$N$$.


 * Quotient Ring: $$\left({R / J, +, \circ}\right)$$ is the quotient ring of a ring $$\left({R, +, \circ}\right)$$ and an ideal $$J$$.


 * Quotient Field: $$\left({F, +, \circ}\right)$$ is a quotient field of an integral domain $$\left({D, +, \circ}\right)$$ iff $$\left({F, +, \circ}\right)$$ contains $\left({D, +, \circ}\right)$ algebraically such that $$\forall z \in F: \exists x \in D, y \in D^*: z = \frac x y$$ where $$\frac x y$$ is $x$ divided by $y$.

Topology
Let $$\left({X, \vartheta}\right)$$ be a topological space.

Let $$\mathcal R \subseteq X^2$$ be an equivalence relation on $$X$$.

Let $$q_\mathcal R: X \to X / \mathcal R$$ be the quotient mapping induced by $$\mathcal R$$.


 * The Quotient Space is the quotient set $$X / \mathcal R$$ whose topology $$\vartheta_{X / \mathcal R}$$ is defined as $$U \in \vartheta_{X / \mathcal R} \iff q_\mathcal R^{-1} \left({U}\right) \in \vartheta \ $$.


 * The Quotient Topology on $$X / \mathcal R$$ by $$q_\mathcal R$$ is the topology $$\vartheta_{X / \mathcal R}$$, also called the identification topology.