# Definition:Real Interval/Unit Interval/Closed

< Definition:Real Interval | Unit Interval(Redirected from Definition:Closed Unit Interval)

## Definition

The closed interval from $0$ to $1$ is denoted $\mathbb I$ (or a variant) by some authors:

- $\mathbb I := \left [{0 \,.\,.\, 1} \right] = \left\{{x \in \R: 0 \le x \le 1}\right\}$

This is often referred to as the **closed unit interval**.

## Also denoted as

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 italic font $I$.

## 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:

\(\displaystyle \openint a b\) | \(:=\) | \(\displaystyle \set {x \in \R: a < x < b}\) | $\quad$ Open Real Interval | $\quad$ | |||||||||

\(\displaystyle \hointr a b\) | \(:=\) | \(\displaystyle \set {x \in \R: a \le x < b}\) | $\quad$ Half-Open (to the right) Real Interval | $\quad$ | |||||||||

\(\displaystyle \hointl a b\) | \(:=\) | \(\displaystyle \set {x \in \R: a < x \le b}\) | $\quad$ Half-Open (to the left) Real Interval | $\quad$ | |||||||||

\(\displaystyle \closedint a b\) | \(:=\) | \(\displaystyle \set {x \in \R: a \le x \le b}\) | $\quad$ Closed Real Interval | $\quad$ |

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

## Sources

- 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Introduction: Special Symbols