Non-Trivial Connected Set in T1 Space is Dense-in-itself

Theorem
Let $T = \left({X, \vartheta}\right)$ be a $T_1$ (Fréchet) topological space which is ultraconnected.

Let $H \subseteq X$ be connected in $T$.

If $H$ has more than one element, then $H$ is dense-in-itself.