TITLE: How to define a "metric" whose range is not the reals?
QUESTION [6 upvotes]: This may sound a very stupid question. Why do we need to restrict a metric from a general set $X$ to map to the positive real numbers? I try to be clearer.
We are given a set $X$ and a totally ordered set ($Y,\succeq $) with least element $0$ and "an addition-like operation on it" denoted by $+$. A metric $d$ is a function $d:X\times X \rightarrow Y $ satisfying the following axioms $\forall x,y,z\in X$: 
(1) $d(x,y)\succeq 0$ if $x\neq y$ and $d(x,y)=0$ if $x=y$;
(2) $d(x,y)=d(y,x)$; (3) $d(x,z)\preceq d(x,y)+d(y,z)$.
Does this definition make any sense? If yes, has work been done on this subject?
Thank you

REPLY [2 votes]: A very general answer in the non symmetric case is given by F.W. Lawvere in the paper 
available for download here entitled "Metric spaces, generalized logic and closed categories".