Definition:Real Interval/Unit Interval/Open

Definition

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

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

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 $J$.

The notation used on this site to denote a real interval is a fairly recent innovation, and was introduced by Niklaus Emil Wirth:

$\ds \openint a b$

$:=$

$\ds \set {x \in \R: a < x < b}$

$\ds \hointr a b$

$:=$

$\ds \set {x \in \R: a \le x < b}$

$\ds \hointl a b$

$:=$

$\ds \set {x \in \R: a < x \le b}$

$\ds \closedint a b$

$:=$

$\ds \set {x \in \R: a \le x \le b}$

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