Definition:Ordering on Extended Real Numbers
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Definition
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
- Ordering on Extended Real Numbers is Ordering
- Ordering on Extended Real Numbers is Total Ordering
- Positive Infinity is Maximal
- Negative Infinity is Minimal
- Results about the usual ordering on the extended real numbers can be found here.