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.
Sign up using Google
Sign up using Facebook
Sign up using Stack Exchange
5 months ago