Generalization of metric that can induce ordering I was wondering if there is some generalization of the concept metric to take  positive and negative and zero values, such that it can induce an order on the metric space? If there already exists such a concept, what is its name?
For example on $\forall x,y \in \mathbb R$, we can use difference $x-y$ as such generalization of metric.
Thanks and regards!

How about a distance defined as relative to an origin? This origin being a point on 1d spaces.
For higher dimensional spaces I'm not sure what would be more interesting, either a finite set of n points or a n-1 dimensional subspace given an n dimensional space (as in 2 points or a line, for 2d planes, a plane for 3d spaces and so on.)
I'm actually interested in such a metric on a finite space: I'm trying to implement Kademlia which defines distance as a XOR on the set of 160 bit long symbols and would like to use ordered sublists to manage the buckets ;)
