There is a folklore correspondence between topology as semi-decidability amongst computer scientists, which is explained in places like:
 
1. [The monograph Synthetic Topology: of Data Types and Classical Spaces](https://www.sciencedirect.com/science/article/pii/S1571066104051357)
2. [An answer by Andrej Bauer on reversing the order of quantifiers](https://cstheory.stackexchange.com/questions/4473/techniques-for-reversing-the-order-of-quanti%ef%ac%81ers/4478#4478)
.

I recently also learnt that [Coarse structures](https://en.wikipedia.org/wiki/Coarse_structure) are a type of "dual" to topologies, in that they capture "global" behaviour versus "local" behvaiour that topologies capture. This viewpoint is explained in:

- [This answer on MathOverflow to the question "dualizing topology"](https://mathoverflow.net/a/346748/123769), which writes down the topology axioms using category theory and then dualizes the construction.
- In general, the view appears to be held that Coarse Stuctures is the correct way to dualize a topology to study large-scale phenomena. My understanding is that it was used very effectively by Gromov to study hyperbolic groups by considering quasi-isometry.

So, it is natural to ask, "what is the computational equivalent of a coarse structure"? I was hoping the answer would be something like "co-induction" / "making progress", since topology seems to be about "deciding" things. However, I have no clue how to _proceed_ with such a question. I'd love some insight into this.