# Does the notion of “a family of compact sets” make sense?

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.

• You certainly don't have uniqueness : for example, the discrete topology and the cofinite topology have both finite sets as compact. But maybe starting from a family of compact (you should probably ask stability by intersections) you can create a topology. The naive way would be to take your topology to be the complement of the compact family. – user171326 Apr 26 '17 at 19:22
• This is related to the notion of a compactly generated space. – Nate Eldredge Apr 26 '17 at 19:23
• @N.H.: In the cofinite topology, every set is compact, so that example doesn't work. – Nate Eldredge Apr 26 '17 at 19:24
• Of course you're right. – user171326 Apr 26 '17 at 19:27
• On the other hand, we can take a topology on $\mathbb R$ where opens are the complements of countable set (plus the empty set). This is a topology on $\mathbb R$ and this time if I'm not mistaken the only compact set are the finite set. – user171326 Apr 26 '17 at 19:29