In the definition of metric space, only one thing strikes me as unnatural: the requirement that $d(x,y) = 0$ implies $x = y$. As a programmer, I don't find it uncommon to deal with equivalence relations (or, more generally, preorders) whose equivalence classes don't have computable canonical representatives. If I wanted to make a metric space out of these, it would be very natural to require that $d(x,y) = 0$ whenever $x$ and $y$ belong in the same equivalence class.
So let's say I change the definition of a metric space $(M,d)$, so that only the following axioms are required:
- $d(x,x) = 0$
- $d(x,y) = d(y,x) \ge 0$
- $d(x,y) + d(y,z) \ge d(x,z)$
What I want to understand is the effect of this change on the definition of “metrizable topological space”. Clearly, some non-metrizable spaces under the old definition become metrizable under the new one, e.g., spaces with the indiscrete topology. I'd like to know...
- Do any spaces remain non-metrizable?
- Do any interesting / useful / naturally arising spaces remain non-metrizable?
- Could you provide examples?