# Category:Alexandroff Extensions

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This category contains results about **Alexandroff Extensions**.

Definitions specific to this category can be found in Definitions/Alexandroff Extensions.

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 $S$.

## Source of Name

This entry was named for Pavel Sergeyevich Alexandrov.

## Pages in category "Alexandroff Extensions"

The following 12 pages are in this category, out of 12 total.

### A

- Alexandroff Extension is Compact
- Alexandroff Extension is Topology
- Alexandroff Extension of Rational Number Space is Biconnected
- Alexandroff Extension of Rational Number Space is Connected
- Alexandroff Extension of Rational Number Space is not Hausdorff
- Alexandroff Extension of Rational Number Space is Sequentially Compact
- Alexandroff Extension of Rational Number Space is T1 Space
- Alexandroff Extension which is T2 Space is also T4 Space