# Definition:Real Interval/Closed

## Definition

Let $a, b \in \R$.

The closed (real) interval from $a$ to $b$ is defined as:

$\closedint a b = \set {x \in \R: a \le x \le b}$

## Notation

An arbitrary (real) interval is frequently denoted $\mathbb I$.

Sources which use the $\textbf {boldface}$ font for the number sets $\N, \Z, \Q, \R, \C$ tend also to use $\mathbf I$ for this entity.

Some sources merely use the ordinary $\textit {italic}$ font $I$.

Some sources prefer to use $J$.

### Wirth Interval Notation

The notation used on this site to denote a real interval is a fairly recent innovation, and was introduced by Niklaus Emil Wirth:

 $\ds \openint a b$ $:=$ $\ds \set {x \in \R: a < x < b}$ Open Real Interval $\ds \hointr a b$ $:=$ $\ds \set {x \in \R: a \le x < b}$ Half-Open (to the right) Real Interval $\ds \hointl a b$ $:=$ $\ds \set {x \in \R: a < x \le b}$ Half-Open (to the left) Real Interval $\ds \closedint a b$ $:=$ $\ds \set {x \in \R: a \le x \le b}$ Closed Real Interval

The term Wirth interval notation has consequently been coined by $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also known as

A closed real interval can also be referred to as compact.

Some sources do not explicitly define an open interval, and merely to a closed real interval as an interval.

Such imprecise practice is usually discouraged.

## Examples

### Example $1$

Let $I$ be the closed real interval defined as:

$I := \closedint 1 3$

Then $3 \in I$.

## Technical Note

The $\LaTeX$ code for $\closedint {a} {b}$ is \closedint {a} {b} .

This is a custom $\mathsf{Pr} \infty \mathsf{fWiki}$ command designed to implement Wirth interval notation.