Question on Closure of Relatively Closed Set. If $A \subset \mathbb{R}^{n}$ is closed, then
$\overline{A} \cap \mathbb{R}^{n} = A$,
where $ \overline{ \vphantom{A}\cdot\vphantom{A} }$ denotes the closure operator in $\mathbb{R}^{n}$.
My question is:
If $A$ is relatively closed in $X \subset \mathbb{R}^{n}$, does it then hold that
$\overline{A} \cap X = A$?
Edit: From this entry in the Encyclopedia of Mathematics, it seems that the second equation is in fact equivalent to $A$ being relatively closed in $X$.

Yes it is, $A$ is relatively closed in $X$ so it exists $C$ closed subset in $E$ such as $A = C \cap X$.
$\overline{C \cap X} \subset \bar C\cap \bar X$.
Consequently : $\bar A \cap X = \overline{C \cap X} \cap X \subset \bar C\cap \bar X \cap X = C\cap X = A$
Since $A \subset X$ and $A \subset \bar A$ the other inclusion is obvious.
Finally $$\bar A \cap X = A$$
