Open Extension Space is Path-Connected

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

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

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

Proof

 * Open Extension Space is Ultraconnected
 * Ultraconnected Space is Path-Connected