# Definition:Real Interval/Unbounded Open

## Definition

There are two unbounded open intervals involving a real number $a \in \R$, defined as:

 $\ds \openint a \to$ $:=$ $\ds \set {x \in \R: a < x}$ $\ds \openint \gets a$ $:=$ $\ds \set {x \in \R: x < a}$

Using the same symbology, the set $\R$ can be represented as an unbounded open interval with no endpoints:

$\openint \gets \to = \R$

## 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 denoted as

The notation using $\infty$ is usual:

 $\ds \openint a \infty$ $:=$ $\ds \set {x \in \R: a < x}$ $\ds \openint {-\infty} a$ $:=$ $\ds \set {x \in \R: x < a}$ $\ds \openint {-\infty} \infty$ $:=$ $\ds \R$

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the $\gets \cdots \to$ notation is preferred.

## Also known as

Some sources refer to these as open infinite (real) intervals.