Sierpiński Space is Ultraconnected

Theorem

Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.

Then $T$ is ultraconnected.

Proof

The only closed sets of $T$ are $\O, \set 1$ and $\set {0, 1}$.

$\set 1$ and $\set {0, 1}$ are not disjoint.

Hence the result by definition of ultraconnected.

$\blacksquare$

Also see

• Particular Point Space is not Ultraconnected

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $11$. Sierpinski Space: $18$