Let's say I have a set $X$ and a set $O$ and I am able to define a distance function $d(x,o)\in\mathbb R, x\in X, o\in O$. The only condition I place on $d$ is $d(x,o)>0$. So obviously it is no a metric, since metric calculates distances between elements of the same set and also has some more conditions it must fulfill. Also, I think, it is not a premetric, because $x$ and $o$ are from different sets. Is there a name for such a function or is distance function the correct (and only) name?
Background: In my case, in one application $o\subset X$, but in some applications $o$ is something completely different.