# Definition:Real Interval/Open

## Definition

Let $a, b \in \R$.

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

$\openint a b := \set {x \in \R: a < x < b}$

## 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 $I$.

### 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:

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

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

## Examples

### Example $1$

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

$I := \openint 1 2$

Then $3 \notin I$.

### Example $2$

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

$I := \openint 0 2$

Then $2 \notin I$.

## Technical Note

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

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