[gowld]

The common metric on the reals one of only two that exist as completions of Q!

The other is the p-padics (basically, low-order bits matter more than high-order bits), which have distance but not ordering.

[gottheUIblues]

There are infinitely many different p-adic completions of the rationals for each prime p, so we have 2-adics, 3-adics, 5-adics etc, all different.