Definition:Extended Real Number Line

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 defined as
Some treatises define $\overline \R$ without the negative infinity $-\infty$, the Alexandroff extension of $\R$, isomorphic to the topological group of complex numbers with norm $1$ under multiplication..

This has the benefit that extended real addition is defined on all of $\overline \R$.

A drawback is that not all suprema and infima exist.

Depending on the context one may decide which form is most suitable.

Also known as
This structure can be referred to as the extended real line or the extended (set of) real numbers.

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

Also see

 * Equivalence of Definitions of Extended Real Number Line
 * Definition:Neighborhood of Infinity (Real Analysis)
 * Definition:Alexandroff Extension