# Definition:Real Interval/Unit Interval/Closed

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## Definition

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

- $\mathbb I := \closedint 0 1 = \set {x \in \R: 0 \le x \le 1}$

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

## 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Sources

- 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets - 1965: Claude Berge and A. Ghouila-Houri:
*Programming, Games and Transportation Networks*... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Introduction: Special Symbols