This is inspired by a recent math.SE question.

Given that mathematicians like to come up with theoretical constructs which do not necessarily always have any practical purpose (but sometimes provide lots of fun anyway): to generalize the above question, let's define a "distance function" on reals greater than $1$ by $$d(x,y):=\inf_\limits{m,n\in\mathbb N}|x^m-y^n|.$$ As $0\not\in\mathbb N$, this function is supposedly rather pathological. I think that certain values of $d(x,y)>0$ can at least be found as minima, e.g. if $x=\frac{\sqrt{5}+1}2$, then $x^n$ is close to a Lucas number, so if $y$ then is an integer (or a $k$th root of an integer), the question of finding $d(x,y)$ essentially boils down to the question whether Lucas numbers can be infinitely often odd powers (which is very probably not the case).

Maybe there are any other fun facts that can be said about the function $d$, starting with the question:

Is it possible that $d$ is (against intuition) continuous in each variable?

Note that one could also ask:

Does it satisfy the triangle inequality?

Now for that, we would obviously need to have $d(x,y)=d(x^k,y)=d(x,y^k)$ for all $x,y>1$ and all $k\in\mathbb N$, but as soon as there is a $d(x,y)>0$ given by a minimum (as probably for the case above), this fails.

Any other ideas?