Let $X$ be a compact connected metric space.
Let $a,b$ be distinct points in $X$.
How do I prove that there exists a countable dense subset $D$ of $X$ which contains neither $a$ nor $b$?
Since $X$ is a compact metric space, it is obviously separable. However, I think connectedness should be used to derive that there exists such $D$ not containing two points. How do I prove it ?