Definition:Ordering on Extended Real Numbers

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

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


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

and demanding that $\le$ remains antisymmetric.

The antisymmetry prevents undesirable possibilities like $+\infty \le -\infty$ from occurring.

Also see

 * Ordering on Extended Real Numbers is Ordering
 * Ordering on Extended Real Numbers is Total Ordering