# Definition:Quotient

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## Disambiguation

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Quotient may refer to:

### Arithmetic

Let $/$ denote the operation of Division on a standard number field $\Q$, $\R$ or $\C$.

Let $q = p / d$.

Then $q$ is the quotient of $p$ (divided) by $q$.

### Algebra

• The quotient of $a$ on division by $b$ is the unique number $q$ such that $a = q b + r, 0 \le r < \size b$ (see the Division Theorem).

### Abstract Algebra

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

$\forall z \in F: \exists x \in D, y \in D^*: z = \dfrac x y$
where $\dfrac x y$ is $x$ divided by $y$.

### Topology

Let $\struct {X, \tau}$ be a topological space.

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

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

• The Quotient Space is the quotient set $X / \RR$ whose topology $\tau_{X / \RR}$ is defined as $U \in \tau_{X / \RR} \iff q_\RR^{-1} \sqbrk U \in \tau$.