For $\displaystyle x,y\in\mathbb{R}$, say that $\displaystyle x\sim y$ if $\displaystyle |x-y|\in\mathbb{Q}$. Define $\displaystyle [x]=\{y: y\sim x\}$. Let $\displaystyle X=\{[x]: x\in\mathbb{R}\}$.

If I want $\displaystyle (X,d)$ to be a metric space, is there a non-trivial metric $\displaystyle d$ I can use that will make $\displaystyle X$ complete? If not, what metric would make the most sense to use on this metric space?

$\displaystyle d([x],[y])=???$

Any suggestions?

Note: I'm asking this out of intellectual curiosity. It is not a homework problem.