Definition:Closed Extension Topology

Definition

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

Let $p$ be a point which is not in $S$.

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

Let $\tau^*_p$ be the set defined as:

$\tau^*_p = \left\{{U \cup \left\{{p}\right\}: U \in \tau}\right\} \cup \left\{{\varnothing}\right\}$

That is, $\tau^*_p$ is the set of all sets formed by adding $p$ to all the open sets of $\tau$ and including the empty set.

Then $\tau^*_p$ is the closed extension topology of $\tau$, and $T^*_p = \left({S^*_p, \tau^*_p}\right)$ is the closed extension space of $T = \left({S, \tau}\right)$.

Also see

• Closed Extension Topology is a Topology
• Closed Sets of Closed Extension Topology (which explains the name closed extension topology).
• Open Exension Topology

• Results about closed extension topologies can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{II}: \ 12: \ 20$