Definition:Alexandroff Extension of Real Number Line
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The Alexandroff extension of the real number line $\R^*$ is defined as:
- $\R^* := \R \cup \set \infty$
- 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.
- 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