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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 $\mathbb{R}^n.$

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    Maybe I'm misunderstanding the question, but how about $n=1$, $Y=(0,1)$, and $X=[1/2,1)$? You might ask the same question, but demand that the boundary of $Y$ be disjoint from the boundary of $X$.2012-09-11

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Hint:

If a set $A$ is not closed in $\mathbb{R}^n$, then there is a sequence $x_n\in A$ converging to $x\in \mathbb{R}^n\setminus A$. (Why?)

Do you see where to go from there?