Topology and measure theory are two examples of fields which include, in their objects of study, sets which are given structure by equipping them with a lattice of a certain type sitting in their power set. Are there other examples of this pattern?
2026-03-25 13:59:45.1774447185
Sets equipped with sublattice of their power sets
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A bornology is an example too.
A convexity structure is not a lattice (not closed under unions, just under intersections and directed unions), but does have the "feel" of a topological structure, to me at least.
A convergence space is a set $X$ and a relation on the set of ultrafilters on it times $X$, etc. So such "second order" (or higher order) structures are not uncommon. Not always lattice based, though.