Definition:Alexandroff Extension of Real Number Line

Definition

The Alexandroff extension of the real number line $\R^*$ is defined as:

$\R^* := \R \cup \set \infty$

that is, the set of real numbers together with an element $\infty$ which is not in $\R$.

Also see

• Definition:Alexandroff Extension for a definition of this concept with respect to the general topological space.

• Results about Alexandroff extensions can be found here.

Source of Name

This entry was named for Pavel Sergeyevich Alexandrov.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions: Exercise $2$
• 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control: $2.2$: Extended-valued functions, semicontinuity