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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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Your accept rate is a bit low, why? –  AD. Sep 11 '12 at 19:34
    
how do I accept if people answer via comments or if the comments were more helpful than the answers? –  Vivek Sep 11 '12 at 19:34
    
There is a checkbox to the left of the answers. –  AD. Sep 11 '12 at 19:38
    
@Vivek Ask the commenter to post the comment as a answer and accept it. –  Ayman Hourieh Sep 11 '12 at 19:38
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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 $X$. –  David Mitra Sep 11 '12 at 19:54
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1 Answer

up vote 2 down vote accepted

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?

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thanks I can work it out now –  Vivek Sep 11 '12 at 19:52
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