Any topology (family of open sets) on a given space determines a family of compact sets on the space (namely, all subsets of the space satisfying the definition of compactness).
Question: Can we go in the opposite direction, i.e. start with a certain family of subsets of the space which are said to be compact?
This would entail at least two sub-questions:
(a) Can a family of compact sets be described axiomatically without referencing a notion of open sets? (I.e. a family of compact sets "for some topology"?)
(b) Given a family of compact sets for some topology, is there only one topology such that this family of compact sets is induced by it? I.e. do we have a well-defined map from families of compact sets to families of open sets?
Motivation: Such a point of view, if possible, might be slightly useful
(1) for motivating proper maps as the analog of continuous maps
(2) in introductory functional analysis, in particular, for making explicit the duality that finer topologies/"more" open sets correspond to "fewer" compact sets, and "more" compact sets correspond to coarser topologies/"fewer" open sets.
In short I think it might be an interesting curiosity, but otherwise don't see any "real" applications.