Excluded Point Space is Path-Connected

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

Then $T^*_{\bar p}$ is path-connected.

Proof 1

 * Excluded Point Topology is Open Extension Topology of Discrete Topology
 * Open Extension Space is Path-Connected

Proof 2

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