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Suppose that $X$ is Hausdorff and $x \in cl(C)$, where $C$ is countable discrete in $X$. Does there exist a sequence $\{x_n\}$ of $C$ converging to $x$?

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No. For example, let $X$ be the Stone–Čech compactification of $\Bbb{N}$, and $C$ be the embedded copy of $\Bbb{N}$. Then $C$ is countable and discrete, and its closure is all of $X$, but no sequence in $C$ converges to a point of $X \setminus C$.

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Hint: What does it mean for $x\in cl(C)$?

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Please say more. –  Paul Dec 23 '12 at 12:23

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