Irreducible Space is not necessarily Path-Connected

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

Then $T$ is not necessarily path-connected.


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.


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