§ Coarse structures
A coarse structure on the set is a collection of relations on :
(called as controlled sets / entourages)
The sets that are controlled are "small" sets.
The bounded coarse structure on a metric space is the set of all relations
such that there exists a uniform bound such that all related elements are within
that bounded distance.
- Closed under subsets: .
- Closed under transpose: if then .
- Closed under finite unions.
- Closed under composition: , where is composition of relations.
We can check that the functions:
are coarse inverses to each other.
I am interested in this because if topology is related to semidecidability,
then coarse structures (which are their dual) are related to..?