What if you order the absolute values of the differences from the midpoint of the interval, where the midpoint is supremum - infimum if bounded, or zero if unbounded? Obviously this fails for closed/open sets like [a,infinity), but wouldn't it work for fully open sets?
Ultimately, the point becomes moot because we focus too much on "normal" sets. Such a definition does nothing to help us answer questions like, what is the least element of the irrationals?