Let $S \subseteq \mathbb{R}^2$ and the distance of any two points in $S$ is rational. Prove $S$ is a countable set.

Question

$S$ is a subset of the $\mathbb{R}^2$ plane, how do we prove that its countable? isn't it trivial that $S$ will be countable if distance is rational, or is $\mathbb{R}^2$ itself a countable set, which is why $S$ is countable? If so, how do we prove $\mathbb{R}^2$ is a countable set?

Answer 1

No, it is not trivial, and of course we know that $\Bbb R^2$ is not countable.

HINT: Let $p$ and $q$ be distinct points of $S$. For each $x\in S$ we know that $d(p,x)$ and $d(q,x)$ are rational, so $x$ lies on a circle of rational radius centred at $p$ and on a circle of rational radius centred at $q$.

• How many circles of rational radius centred at $p$ are there? How many centred at $q$?
• If $C_0$ and $C_1$ are circles with different centres, what is the largest possible cardinality of $C_0\cap C_1$?

Added: To see why it’s not trivial, note that if we gave $\Bbb R^2$ the discrete metric instead of the usual one, then $d(x,y)$ would be rational for every $x,y\in\Bbb R^2$: in that case $d(x,y)=0$ if $x=y$, and $d(x,y)=1$ if $x\ne y$. Thus, in that case $S$ could be any subset of $\Bbb R^2$, countable or uncountable.