If $A$ countable then $\Bbb{R}^2\setminus A$ is path connected

Question

Let $A\subset\Bbb{R}^2$ be countable. I need to prove that $\Bbb{R}^2\setminus A$ is path connected.

I know that through each of $\Bbb{R}^2\setminus A$, there pare uncountably many straight lines, and as there are only countably many points in $A$, uncountably many of these lines will not contain any point of $A$. But why do I construct a path between any two points.

Also can this result be generalised, so that:

If $X$ is uncountable and $A$ is a countable subset of $X^2$, then should $X^2\setminus A$ be path connected.? (where $X$ and $X^2$ are path connected of course)

Answer 1

Let a,b be two points on the plain and not in A.
Draw an arc of a circle of radius r with ab as a cord.
There are uncountable many such arcs and as they are pairwise disjoint execept at the endpoints, almost all of them will miss A. Thus a circlular arc from a to b missing A.