Definition:Alexandroff Extension

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Definition

Let $T = \left({S, \tau}\right)$ be a non-empty topological space.

Let $p$ be a new element not in $S$.

Let $S^* := S \cup \left\{{p}\right\}$.


Let $\tau^*$ be the topology on $S^*$ defined such that $U \subseteq S^*$ is open if and only if:

$U$ is an open set of $T$

or

$U$ is the complement in $T^*$ of a closed and compact subset of $T$.


This topology is called the Alexandroff extension on $S$.


Also known as

The Alexandroff extension on $S$ is also known as:

the Alexandroff compactification topology on $S$.
the one point (or one-point) compactification topology on $S$.


Also see

  • Results about Alexandroff extensions can be found here.


Source of Name

This entry was named for Pavel Sergeyevich Alexandrov.


Sources