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I know this is not true but need to find an example of complete metric space $X$ with subset $E$ such that $\overline{bd(E)}$ has non-empty interior.

BMac
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    Perhaps you know of some subset of $\mathbb R$ that is dense but has empty interior. – GEdgar Apr 29 '19 at 20:35
  • Ah, the rationals are dense in $\mathbb{R}$ but any open ball centered at $q \in \mathbb{Q}$ contains irrationals. So do we see $\mathbb{Q}$ as the boundary of the irrationals? – BMac Apr 29 '19 at 20:42
  • No, $\mathbb Q$ is not the boundary of the irrationals. Use the definition of "boundary" to find what that boundary is. – GEdgar Apr 29 '19 at 20:52
  • Oh duh, $\mathbb{R}$ is the boundary, since it's the intersection of the closure of the rationals and the closure of irrationals. And $\mathbb{R}$ is its own closure, which has non-empty interior. – BMac Apr 29 '19 at 20:57

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The boundary of $\mathbb{Q}$ in $\mathbb{R}$ (usual metric) is $\mathbb{R}$. The boundary of an open set (or equivalently a closed set) is nowhere dense.

Henno Brandsma
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