Thread: Equivalence Class

For , say that if . Define . Let .

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



Any suggestions?

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

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.

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 but .

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 .