# Definition:Alexandroff Extension

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## Definition

Let $T = \struct {S, \tau}$ be a non-empty topological space.

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

Let $S^* := S \cup \set p$.

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

- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $34$. One Point Compactification Topology - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Alexandroff compactification**