Hmm. Well it looks like every quasi uniform space is a topology, but it also looks like maybe it will be easier to make a directed* quasi uniform space then a directed topology.

* "directed" meaning non symmetric, by analogy with graphs.

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Oh, hmm. Hmm, maybe we want a topology on ordered pairs of points, with basis \\[ B_r(p,q)= \\{ (x,y) | d (p,x) + d(x,y) + d (y,q) < r + d (p,q)\\}\\]

I'm on my phone so I can't type out the equations to check if that works, and I'm not up to doing it in my head, but i will look at it later.