Closed Set of Ultraconnected Space is Ultraconnected

Theorem
Let $T = \left({S, \tau}\right)$ be an ultraconnected topological space.

Let $F \subset S$ be a closed set in $T$.

Then $F$ is ultraconnected.

Also see

 * Space is Ultraconnected iff Closed Subsets are Connected
 * Open Subset of Irreducible Space is Irreducible, whose proof is almost the same