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

$\blacksquare$

Proof 2

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

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

$\blacksquare$

Proof 3

Excluded Point Space is Ultraconnected

Ultraconnected Space is Path-Connected

Path-Connected Space is Connected

$\blacksquare$