Irreducible Space is not necessarily Path-Connected
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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.
$\blacksquare$
Also see
- Finite Irreducible Space is Path-Connected
- Irreducible Space with Finitely Many Open Sets is Path-Connected
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness