I have the following result along with the proof. I want to generalize this to $n$-dimension.
Result Let $A \subseteq \mathbb{R}^{2}$ be countable. Then $\mathbb{R}^{2}-A$ is path connected.
Proof. Let $x, y \in \mathbb{R}^{2}-A .$ since there are only countably many points in $A,$ there are only countably many lines through $x$ that hit $A .$ similarly, there are only countably many lines through $y$ that hit $A .$ Therefore there are at least two lines through $x$ that do not hit $A$ and one line through $y$ that does not hit $A$. (Actually, there are uncountably many, but this is all we need.) Then the line through $y$ must intersect one of those two lines through $x$ in a point $z .$ We can then form a path from $x$ to $y$ in $\mathbb{R}^{2}-A$ by taking the union of the line segment from $x$ to $z$ with the line segment from $z$ to $y$ and appropriately parameterizing this geometric construction to form a continuous function $f:[0,1] \rightarrow \mathbb{R}^{2}$ with $f(0)=x$ and $f(1)=y$.
Now can I generalize this proof? i.e. I want to show that $A \subseteq \mathbb{R}^{n}$ be countable, then also $\mathbb{R}^{n}-A$ is path-connected.