Set with Dispersion Point is Biconnected

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

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

Let $p \in H$ be a dispersion point of $H$.

Then $H$ is biconnected.

Proof
Assume the contrary. $H = U \cup V$ for disjoint connected $U, V$. Then w.l.o.g. $p \in U$. Hence $V \subset H - \{ p \}$ is disconnected. $\square$