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I am wondering if it is true, and what the proof would go like, that given a metric space $X$ with a countable dense subset $D$, $\ X\setminus D$ is totally disconnected.

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But $\Bbb R$ with any dense subset removed will be totally disconnected. – Stefan Hamcke Feb 10 '14 at 21:47

This is not true. $\mathbb Q^2\subset\mathbb R^2$ is dense, but in fact $\mathbb R^2\setminus\mathbb Q^2$ is path connected.

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What if X contains exactly two non-cut points? – abf Feb 10 '14 at 21:50
What do you mean, non-cut points? – Ian Coley Feb 10 '14 at 23:41

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