Excluded Point Space is Path-Connected

Theorem

Let $T = \struct {S, \tau_{\bar p} }$ 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

$\blacksquare$

Proof 2

Excluded Point Space is Ultraconnected

Ultraconnected Space is Path-Connected

$\blacksquare$