# Definition:Real Interval/Unit Interval

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## Definition

A **unit interval** is a real interval whose endpoints are $0$ and $1$:

\(\displaystyle \openint 0 1\) | \(:=\) | \(\displaystyle \set {x \in \R: 0 < x < 1}\) | |||||||||||

\(\displaystyle \hointr 0 1\) | \(:=\) | \(\displaystyle \set {x \in \R: 0 \le x < 1}\) | |||||||||||

\(\displaystyle \hointl 0 1\) | \(:=\) | \(\displaystyle \set {x \in \R: 0 < x \le 1}\) | |||||||||||

\(\displaystyle \closedint 0 1\) | \(:=\) | \(\displaystyle \set {x \in \R: 0 \le x \le 1}\) |

### Closed

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**.

### Open

The open interval between $0$ and $1$ is referred to as the **open unit interval**.

- $\left({0 \,.\,.\, 1}\right) = \left\{ {x \in \R: 0 < x < 1}\right\}$

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