Definition:Alexandroff Extension
Jump to navigation
Jump to search
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): Alexandroff compactification