Definition:Real Interval/Unbounded Open

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


 * $\left ({a \,.\,.\, \to} \right) = \left\{{x \in \R: a < x}\right\}$


 * $\left ({\gets \,.\,.\, a} \right) = \left\{{x \in \R: x < a}\right\}$

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


 * $\left ({\gets \,.\,.\, \to} \right) = \R$

Also denoted as
The notation using $\infty$ is usual:
 * $\left ({a \,.\,.\, \infty} \right) = \left\{{x \in \R: a < x}\right\}$


 * $\left ({-\infty \,.\,.\, a} \right) = \left\{{x \in \R: x < a}\right\}$


 * $\left ({-\infty \,.\,.\, \infty} \right) = \R$

On the $\gets \cdots \to$ notation is preferred.

Also see

 * Definition:Open Real Interval
 * Definition:Closed Real Interval
 * Definition:Half-Open Real Interval
 * Definition:Unbounded Closed Real Interval