# 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.

Hence the result.

$\blacksquare$