Open Extension Space is Connected

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

Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.

Then $T^*_{\bar p}$ is a connected space.

Proof
The only open set of $T$ which contains $p$ is $S^*_p$.

Therefore it is impossible to set up a separation of $T$, as $S^*_p$ will always need to be an element of such a separation.