Closure of Subset in Subspace

Theorem
Let $T = \struct{S, \tau}$ be a topological space.

Let $H \subseteq S$ be an arbitrary subset of $S$.

Let $T_H = \struct {H, \tau_H}$ be the topological subspace on $H$.

Let $A \subseteq H$ be an arbitrary subset of $H$.

Then:
 * $\map {\cl_H} A = H \cap \map \cl A$

where:
 * $\map {\cl_H} A$ denotes the closure of $A$ in $T_H$
 * $\map \cl A$ denotes the closure of $A$ in $T$.