Definition:Ordering on Extended Real Numbers

Definition
Let $\overline{\R}$ denote the extended real numbers.

Extend the natural ordering $\le_\R$ on $\R$ to $\overline{\R} = \R \cup \left\{{+\infty, -\infty}\right\}$ 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 $\overline \R \times \overline \R$:


 * ${\le} := {\le_\R} \cup \left\{{ \left({ x, +\infty }\right) }\,\middle\vert\,{ x \in \overline{\mathbb R} }\right\} \cup \left\{{ \left({ -\infty, x }\right) }\,\middle\vert\,{ x \in \overline{\mathbb R} }\right\}$

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

Also see

 * Ordering on Extended Real Numbers is Ordering
 * Ordering on Extended Real Numbers is Total Ordering
 * Positive Infinity is Maximal
 * Negative Infinity is Minimal