Definition:Real Interval/Unbounded Open

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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.

Also see