Let $S \subseteq \mathbb{R}^2$ and the distance of any two points in $S$ is rational. Prove $S$ is a countable set.
132778
All distances are rational prove the set is countable
157513
How can I prove that these sets are countable infinite?
102543
Real definition of "countable set"
41661
Show that the set of all finite subsets of $\mathbb{N}$ is countable.
98151
Suppose that A is a countable set. Show that the set B is also countable if there is an onto function f from A to B.
136006
Prove that any countable cartesian product of countable sets is countable
28391
Prove the compactness of a set B
7607
How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.
85895
Proving all rational numbers including negatives are countable
78763
Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$.
110635
Proving $\mathbb{N}^k$ is countable
86685
Example of a set $S$ that is countable, but the set of limit points is uncountable
154474
Image of countable and uncountable sets
3708
How can I prove that compact subset iff sequentially compact in second countable topological spaces?
68430
Proving the countability of algebraic numbers
128437
How to prove that a finite set is countable
12472
Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable