Let $A$ be a countable subset of $\mathbb{R}^n$, where $n > 1$.
Prove that $\mathbb{R}^n \setminus A$ is path-connected.
Analogy of R^2. Notice that n = 1 does't work.
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for any two points x, y not in A, pick any single parameter family of curves that connects x and y, having only x, y as their common points. There must be one curve c in the family such that c and A have no common points. Otherwise, suppose each curve intersects with A, A will have so many points that are not countable.