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!

If we have a function $\delta$ such that


*

*$\delta(x,y)=0$ if and only if $x=y$

*$\delta(x,y) = -\delta(y,x)$

*If $\delta(x,y)\geq 0$ and $\delta(y,z)\geq 0$, then $0\leq \delta(x,z) \leq \delta(x,y)+\delta(y,z)$.


then $d(x,y) = |\delta(x,y)|$ clearly defines a metric. Furthermore $x\geq y$ if and only if $\delta(x,y)\geq 0$ defines a total order.
Conversely, if $d$ is a metric and $\geq$ a total order, $\delta(x,y) =\begin{cases} d(x,y) & x\geq y \\ -d(x,y) & x<y \end{cases}$ satisfies all the axioms above.
