Definition:Ordering on Extended Real Numbers

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Let $\overline \R$ denote the extended real numbers.

Extend the natural ordering $\le_\R$ on $\R$ to $\overline \R = \R \cup \set {+\infty, -\infty}$ by imposing:

$\forall x \in \overline \R: -\infty \le x$
$\forall x \in \overline \R: x \le +\infty$

That is, considering the relations $\le$ and $\le_\R$ as subsets of $\overline \R \times \overline \R$:

${\le} := {\le_\R} \cup \set {\tuple {x, +\infty}: x \in \overline \R} \cup \set {\tuple {-\infty, x}: x \in \overline \R}$

where $\tuple {x, +\infty}$ and $\tuple {-\infty, x}$ denote ordered pairs in $\overline \R \times \overline \R$.

The ordering $\le$ is called the (usual) ordering on $\overline \R$.

Also see

  • Results about the usual ordering on the extended real numbers can be found here.