# Finite Irreducible Space is Path-Connected

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## 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.

$\blacksquare$