Excluded Point Space is Connected

Theorem
Let $T = \left({S, \tau_{\bar p}}\right)$ be an excluded point space.

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

Proof 1
We have:
 * Excluded Point Topology is Open Extension Topology of Discrete Topology
 * Open Extension Space is Connected

Proof 2
Alternatively, the same argument can be used as for Open Extension Space is Connected:

The only open set of $T$ which contains $p$ is $S$.

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

Proof 3

 * Excluded Point Space is Ultraconnected
 * Ultraconnected Space is Path-Connected
 * Path-Connected Space is Connected