Definition:Closed Extension Topology

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

Let $p$ be a new element for $S$ such that $S^*_p := S \cup \set p$.

Let $\tau^*_p$ be the set defined as:
 * $\tau^*_p := \set {U \cup \set p: U \in \tau} \cup \set \O$

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:
 * $T^*_p := \struct {S^*_p, \tau^*_p}$ is the closed extension space of $T = \struct {S, \tau}$.

Also see

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