Isolated Point of Closure of Subset is Isolated Point of Subset

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

Let $H \subseteq T$ be a subspace of $T$.

Let $\operatorname{cl} \left({H}\right)$ denote the closure of $H$.

Let $x \in \operatorname{cl} \left({H}\right)$ be an isolated point of $\operatorname{cl} \left({H}\right)$.

Then $x$ is also an isolated point of $H$.