Show that $(R \times R) \setminus A$ is path connected

Question

Let $R$ be the usual topological space and $A$ be a countable subset of $R \times R$.

How to show that $(R \times R) \setminus A$ is path connected?

Answer 1

Let a,b be two points in RxR - A. Within RxR,
there are uncountablely many, pair disjoint
(except at the end points), arcs from a to b.
Almost all will miss countable A.

Arc in the geometric meaning as a connected portion of a circle.