Let Y be an open subset of $\mathbb{R}^n$. If X is a closed subset of Y, disjoint from the boundary of Y, is it true that X is a closed subset of $\mathbb{R}^n$? How do I show this?
Edit: Let X be contained in a closed set B of $\mathbb{R}^n$ which is contained in Y and which is disjoint from the boundary of Y. Then X is closed in $\mathbb{R}^n.$