1. ## Equivalence Class

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.

2. I guess youv'e thought of it, but why not define it as:
d([x],[y])=0 if [x]=[y] and |x-y| if |x-y| isn't rational (which means when [x] is different than [y]).

This seems as a plausible metric for X.

3. Originally Posted by InvisibleMan
why not define it as:
d([x],[y])=0 if [x]=[y] and |x-y| if |x-y| isn't rational (which means when [x] is different than [y]).
This metric is not well defined since we have $\displaystyle [\sqrt{2}]=[\sqrt{2}+\tfrac{1}{2}]$ but $\displaystyle d([\sqrt{2}],[0])\neq d([\sqrt{2}+\tfrac{1}{2}],[0])$.

4. That's the exact problem I'm running into when trying to come up with a metric for this. We need to find some invariant for all $\displaystyle x\in[x]$.