Finite Irreducible Space is Path-Connected

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

Then $T$ is path-connected.

Proof
By Power Set of Finite Set is Finite, the power set $\mathcal P \left({S}\right)$ is finite.

By Subset of Finite Set is Finite, $\tau \subseteq \mathcal P \left({S}\right)$ is finite.

The result follows from Irreducible Space with Finitely Many Open Sets is Path-Connected.

Also see

 * Irreducible Space is not necessarily Path-Connected