Is there any way to show that the set of disjoint translations of the cantor ternary set is countable?
That is show that there are countably many disjoint sets of the form $\{x+C: x\in \mathbb{R}\}$???
Thanks
Is there any way to show that the set of disjoint translations of the cantor ternary set is countable?
That is show that there are countably many disjoint sets of the form $\{x+C: x\in \mathbb{R}\}$???
Thanks
For any $x,y\in\Bbb R$ we have $(x+C)\cap(y+C)\ne\varnothing$ iff $x-y\in C-C=[-1,1]$ iff $|x-y|\le 1$. Suppose that $A\subseteq R$ and $\{a+C:a\in A\}$ is pairwise disjoint; then for distinct $a,b\in A$ we must have $|a-b|>1$, and $A$ must be countable.