# Definition:Real Interval/Unbounded Open

## Definition

There are two unbounded open intervals involving a real number $a \in \R$, defined as:

 $\displaystyle \openint a \to$ $:=$ $\displaystyle \set {x \in \R: a < x}$ $\displaystyle \openint \gets a$ $:=$ $\displaystyle \set {x \in \R: x < a}$

Using the same symbology, the set $\R$ can be represented as an unbounded open interval with no endpoints:

$\openint \gets \to = \R$

## Also denoted as

The notation using $\infty$ is usual:

 $\displaystyle \openint a \infty$ $:=$ $\displaystyle \set {x \in \R: a < x}$ $\displaystyle \openint {-\infty} a$ $:=$ $\displaystyle \set {x \in \R: x < a}$ $\displaystyle \openint {-\infty} \infty$ $:=$ $\displaystyle \R$

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the $\gets \cdots \to$ notation is preferred.