[mrmyers]

Why wouldn't the following ordering work? {(x,y) | (|x| ≤ |y|) & ((|x| = |y| & x ≠ y) ⇒ x < 0 < y)}

[v64]

With your ordering, what is the least element of the open interval (0,1)?

[nerdponx]

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?

[v64]

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?

[mrmyers]

Ah, right. Makes sense.