Proposition 3.13.2.label Let $X$ be a topological space, $Y\suf X$ be a subspace then for any subset $A\suf X$
\begin{align*}\cl{A\cap Y}^{Y}=\cl{A}^{X}\cap Y\end{align*}
In particular if $Y$ is closed in $X$ and $A\suf Y$ then
\begin{align*}\cl{A}^{Y}=\cl{A}^{X}\end{align*}
Proof. Observe
\begin{align*}\cl{A\cap Y}^{Y}=\caps{}{A\cap Y\suf C\suf Y\\ C\text{ closed in }Y}C=\caps{}{A\cap Y\suf C\cap Y\suf Y\\C\text{ closed in }X}C\cap Y=Y\cap \caps{}{A\suf C\\C\text{ closed in }X}C=Y\cap \cl{A}^{X}\end{align*}
$\square$
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