Definition:Real Interval/Empty

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