Definition:Extended Real Number Line

Also defined as
Some sources define $\overline \R$ as $\R \cup \set \infty$, that is, without the negative infinity $-\infty$.

This is the Alexandroff extension of $\R$.

This is 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
 * the extended (set of) real numbers

Also, the notations $\sqbrk {-\infty, +\infty}$ and $\closedint {-\infty} {+\infty}$ can be encountered, extending the notation for real intervals.

Also see

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

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 usual (Euclidean) topology on $\R$
 * A $\sigma$-algebra extending the Borel $\sigma$-algebra $\map \BB \R$ on $\R$