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Given an arbitrary bounded region $ \emptyset \neq \Omega \subset \Bbb R^n$, I would like to find a countable set $A \subset \Omega$ so that $A$ has no limit points in $\Omega$ but every point in the boundary $\partial \Omega$ is a limit point of $A$. Any assistance would be appreciated.

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Let $\Omega \subset \mathbb R^n$ be a bounded open set. Note that $\partial \Omega$ is a compact set. For each $n \in \mathbb N$ cover the boundary of $\Omega$ by balls $\{B_x\}_{x \in \partial \Omega}$ of radius $1/n$. Pick a finite subcover of these balls $B_1,\dots,B_k$ and for each $B_i$ pick an element $x \in B_i \cap \Omega$ and call the union of these points $X_n$. Then let $X$ be the union of the $X_n$, $X$ is a countable union of countable (finite even) sets, so is countable. Then it's not hard to show $X$ has no limit points on the boundary, but for any point $y \in \partial \Omega$ we can construct a sequence of points in $X$ converging to $y$.

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Appreciate it, thanks! –  user59151 Jan 21 '13 at 22:25
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