Closed Set of Ultraconnected Space is Ultraconnected

Theorem
Let $X$ be an ultraconnected topological space.

Let $F\subset X$ be an closed subset.

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