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.

$\blacksquare$


Also see


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
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