Definition:Extended Real Number Line

Definition
The set of extended real numbers $\overline{\R}$ is defined as:


 * $\overline{\R} := \R \cup \left\{{+\infty, -\infty}\right\}$

that is, the set of real numbers together with two auxiliary symbols:


 * $+\infty$, positive infinity
 * $-\infty$, negative infinity

Structures on $\overline{\R}$
$\overline{\R}$ can be endowed with the following structures:


 * An ordering extending the standard ordering on $\R$
 * Extended Real Addition, extending real addition $+$
 * Extended Real Subtraction, extending real subtraction $-$
 * Extended Real Multiplication, extending real multiplication $\cdot$
 * A topology extending the Euclidean topology on $\R$
 * A $\sigma$-algebra extending the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$ on $\R$

Also known as
Some authors call this the extended real line.

Also, the notations $\left[{-\infty, +\infty}\right]$ and $\left[{-\infty \,.\,.\, +\infty}\right]$ can be encountered, mimicking the notation for real intervals.