Metric Closure and Topological Closure of Subset are Equivalent

Theorem
Let $M = \struct{A, d}$ be a metric space.

Let $T = \struct{A, \tau}$ be the topological space with the topology induced by $d$.

Let $H \subseteq A$.

Then:
 * the metric closure of $H$ in $M$ equals the topological closure of $H$ in $T$