If $S$ is a countable subset of $R^2$, show that for any two points $x, y \in R^2 \setminus S$, there is a parallelogram in $R^2\setminus S$ having $x, y$ as opposite vertices.
What can I do for this question? I am totally lost. Please give me some help. Appreciate.
Let $x, y \in R^2 \setminus S$, but there is no such parallelogram.
It means that all parallerograms with $x, y$ as opposite vertices contain at least one point $p \in S$. Of course, $p \ne x$ and $p \ne y$.
The set of such points $p$ must be finite or countable, and it is not possible, because there are uncountable set of parallelograms with only 2 common points, namely $x$ and $y$ (different from every point $p$).