# Definition:Quotient

### 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 $d$.

### 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).

### Quotient Structure

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 $y$.

### Quotient (Topological) Space

Let $T = \struct {S, \tau}$ be a topological space.

Let $\RR \subseteq S \times S$ be an equivalence relation on $S$.

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

Let $\tau_\RR$ be the quotient topology on $S / \RR$ by $q_\RR$:

$\tau_\RR := \set {U \subseteq S / \RR: q_\RR^{-1} \sqbrk U \in \tau}$

The quotient space of $S$ by $\RR$ is the topological space whose points are elements of the quotient set of $\RR$ and whose topology is $\tau_\RR$:

$T_\RR := \struct {S / \RR, \tau_\RR}$

Hence:

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

## Linguistic Note

The word quotient derives from the Latin word meaning how often.