# Definition:Real Interval/Unbounded without Endpoints

## Definition

The **unbounded interval without endpoints** is equal to the set of real numbers:

- $\openint \gets \to = \R$

## 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:

\(\ds \openint {-\infty} \infty\) | \(:=\) | \(\ds \R\) |

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

## Also known as

Some sources refer to this as an **open infinite (real) interval**.

## Also see

- Definition:Open Real Interval
- Definition:Closed Real Interval
- Definition:Half-Open Real Interval
- Definition:Unbounded Closed Real Interval
- Definition:Unbounded Open Real Interval

## Sources

- 1973: G. Stephenson:
*Mathematical Methods for Science Students*(2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.2$ Operations with Real Numbers