Open Extension Space is Path-Connected

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ 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

$\blacksquare$


Sources