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If a set $E$ is contained in $\mathbb R^n$ with the standard euclidean norm and if define another set $B$ as the points in $\mathbb R^n$ whose distance to the set $E$ is zero, is it true that $E=B$?

I think that is obvious if $E$ is a closed set, but the statement is valid if $E$ is an open set?

Thanks for any clue!

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Think about the points on the boundary of the set $E.$ – cats Jan 17 '13 at 1:46
What happens with the open set $E=(0,1)$ contained in $\mathbb{R}$? What is $d(0,E)$? – Sigur Jan 17 '13 at 1:51
Yes. It is the definition of distance. – Tom Jan 17 '13 at 2:19
up vote 1 down vote accepted

The answer is no.

Let $n=1, E=\mathbb R\setminus \{0\}$(the real line minus the origin). ($E$ is open.)

The origin $0$ is not contained in $E$ but the distance from the origin to $E$ is zero. So $B=\mathbb R$.

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You have the follow B=$\bar{E}$

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