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.
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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1 year ago