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If $D$ is a domain(open, connected set in $\mathbb{C}$), is it true that any uncountable subset of $D$ has a limit point in $D$? Obviously, uncountable subset has a limit point in $\mathbb{C}$, but I'm not sure if there is one in $D$.

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up vote 1 down vote accepted

Yes, it’s true for any uncountable subset $D$ of $\Bbb C$.

$\Bbb C$ has a countable base $\mathscr{B}$ of open sets. Let $\mathscr{B}_0=\{B\in\mathscr{B}:B\cap D\text{ is countable}\}$, and let $D_0=D\setminus\bigcup\mathscr{B}_0$. $\bigcup\{D\cap B:B\in\mathscr{B}_0\}$ is a countable union of countable subsets of $D$, so it’s countable, and $D_0$ is therefore uncountable. By construction every open nbhd of every $z\in D_0$ has uncountable intersection with $D_0$ and hence with $D$.

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