# Definition:Open Extension Topology

## Definition

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

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

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

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

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

That is, $\tau^*_{\bar p}$ is the set of all sets formed by taking all the open sets of $\tau$ and adding to them the set $S^*_p$.

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

The open sets of $T^*_{\bar p}$ can be seen to be the same as the open sets of $T$, but with $S^*_p$ added.

## Also see

• Results about open extension topologies can be found here.