Finite Irreducible Space is Path-Connected
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Theorem
Let $T = \struct {S, \tau}$ be a finite irreducible topological space.
Then $T$ is path-connected.
Proof
By Power Set of Finite Set is Finite, the power set $\powerset S$ is finite.
By Subset of Finite Set is Finite, $\tau \subseteq \powerset S$ is finite.
The result follows from Irreducible Space with Finitely Many Open Sets is Path-Connected.
$\blacksquare$