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