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

Theorem
Let $T = \left({S, \tau}\right)$ be a $T_1$ (Fréchet) topological space.

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

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

Proof
Suppose $H$ is not dense-in-itself.

Then $\exists x \in H$ such that $x$ is isolated in $H$.

That is, $\exists U \in \tau: U \cap H = \left\{{x}\right\}$.

Let $D = \left({\left\{{0, 1}\right\}, \phi}\right)$ be the discrete space of two points.

Consider the mapping $f: T \to D$.