Irreducible Space is not necessarily Path-Connected

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

Then $T$ is not necessarily path-connected.

Proof
Let $T$ be a countable finite complement space.

From Finite Complement Space is Irreducible, $T$ is an irreducible space.

From Countable Finite Complement Space is not Path-Connected, $T$ is not path-connected.

Hence the result.

Also see

 * Finite Irreducible Space is Path-Connected