@Matthew: that's an interesting take on the question! But did you check whether each one of your two systems of open balls actually *is* a basis?
Turning a Lawvere metric space into (something like) a topological space in a nice way is known to be quite tricky. I think that there are some well-known answers, but I don't understand the details. There's interesting material along these lines in the papers of Jean Goubault-Larrecq, as in [this one](https://arxiv.org/abs/1606.05445). It's much easier to turn a Lawvere metric space into a [quasiuniform space](https://en.wikipedia.org/wiki/Uniform_space), for those who are into this sort of stuff.