Closure of Subset in Subspace

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

Let $H$ be a subset of $S$.

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

Let $A$ be a subset of $H$.

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

where
 * $\map {\operatorname{cl}_H} A$ denotes the closure of $A$ in $T_H$
 * $\map {\operatorname{cl}} A$ denotes the closure of $A$ in $T$