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Let $(X,d)$ be a metric space and let $A\subset X$. If $A$ is either open or closed, then $(\partial A)^{\circ} = \varnothing$. I am asked to find a metric space and a subset that the interior of its boundary is not empty. I have tried something with the discrete metric, but then realized that since every set in a discrete metric is a union of open sets, every set is open, so this has no future.

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You're right that a discrete metric won't work here. HINT: you're looking for a set with "big" boundary but "small" interior. Work in $\mathbb{R}$ with the usual metric; can you think of a set that "fills out" $\mathbb{R}$, but with lots of "holes"?

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    $\begingroup$ Take $A = \mathbb{Q}$, then $\partial\mathbb{Q} = \mathbb{R}$, and $\mathbb{R}^{\circ} = \mathbb{R}$. Great guidance! $\endgroup$ – Joshhh Nov 21 '16 at 19:59
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Hint: try a countable dense subset of $[0,1]$.

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