Definition:Real Interval/Notation/Unbounded Intervals

Definition

Some authors (sensibly, perhaps) prefer not to use the $\infty$ symbol and instead use $\to$ and $\gets$ for $+\infty$ and $-\infty$ repectively.

In Wirth interval notation, such intervals are written as follows:

$\ds \hointr a \to$

$:=$

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

$\ds \hointl \gets a$

$:=$

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

$\ds \openint a \to$

$:=$

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

$\ds \openint \gets a$

$:=$

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

$\ds \openint \gets \to$

$:=$

$\ds \set {x \in \R} = \R$

$\ds \left [{a, \to} \right]$

$=$

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

$\ds \left [{\gets, a} \right]$

$=$

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

$\ds \left ]{\, a, \to} \right]$

$=$

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

$\ds \left [{\gets, a \,} \right[$

$=$

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