# Definition:Real Interval/Unbounded Closed

## Definition

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

 $\displaystyle \hointr a \to$ $:=$ $\displaystyle \set {x \in \R: a \le x}$ $\displaystyle \hointl \gets a$ $:=$ $\displaystyle \set {x \in \R: x \le a}$

## Also denoted as

The notation using $\infty$ is usual:

 $\displaystyle \hointr a \infty$ $:=$ $\displaystyle \set {x \in \R: a \le x}$ $\displaystyle \hointl {-\infty} a$ $:=$ $\displaystyle \set {x \in \R: x \le a}$

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

## Notation

An arbitrary interval is frequently denoted $\mathbb I$, although some sources use just $I$. Others use $\mathbf I$.

### 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 unbounded closed real interval defined as:

$I := \hointl \gets 3$

Then $2 \in I$.