Proving that $\Bbb R^{n} \setminus S$ is connected if $S$ is countable

Question

Let $n>2$. Let $S$ be a set that is at most countable. Prove, that $\Bbb R^{n} \setminus S$ is a connected set.

Let's start $n=2$.

How to show it in a formal way?

Answer 1

$\mathbb R^n\setminus S$ is even pathwise connected. Let $a,b\in \mathbb R^n\setminus S$ and let $\ell\subset \mathbb R^n$ be a line with $a,b\notin\ell$. Then for $c,c'\in\ell$ with $c\ne c'$ the line segments $ac$ and $ac'$ intersect only in $a$, and the line segments $bc$ and $bc'$ intersect only in $b$. Hence for at most countably many $c\in\ell$, $ac\cup bc$ intersects $S$. Since $\ell$ has more than countably many points, for suitable $c\in\ell$ we obtain a path from $a$ to $b$ in $\mathbb R^n\setminus S$.