# Definition:Real Interval/Empty

## Definition

Let $a, b \in \R$.

Let $\left [{a \,.\,.\, b} \right]$, $\left [{a \,.\,.\, b} \right)$, $\left ({a \,.\,.\, b} \right)$ and $\left ({a \,.\,.\, b} \right)$ be real intervals: closed, half-open and open as defined.

When $a > b$:

 $\ds \closedint a b$ $=$ $\, \ds \set {x \in \R: a \le x \le b} \,$ $\, \ds = \,$ $\ds \O$ $\ds \hointr a b$ $=$ $\, \ds \set {x \in \R: a \le x < b} \,$ $\, \ds = \,$ $\ds \O$ $\ds \hointl a b$ $=$ $\, \ds \set {x \in \R: a < x \le b} \,$ $\, \ds = \,$ $\ds \O$ $\ds \openint a b$ $=$ $\, \ds \set {x \in \R: a < x < b} \,$ $\, \ds = \,$ $\ds \O$

When $a = b$:

 $\ds \hointr a b \ \$ $\, \ds = \,$ $\ds \hointr a a$ $=$ $\, \ds \set {x \in \R: a \le x < a} \,$ $\, \ds = \,$ $\ds \O$ $\ds \hointl a b \ \$ $\, \ds = \,$ $\ds \hointl a a$ $=$ $\, \ds \set {x \in \R: a < x \le a} \,$ $\, \ds = \,$ $\ds \O$ $\ds \openint a b \ \$ $\, \ds = \,$ $\ds \openint a a$ $=$ $\, \ds \set {x \in \R: a < x < a} \,$ $\, \ds = \,$ $\ds \O$

Such empty sets are referred to as empty intervals.

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