Inverse Closure Problem

In the undergraduate complexity course we spend some time on closure properties such as (1) REG closed under UNION, INTER, COMP and (2) R.E. closed under UNION and INTER but NOT COMP.

I propose the following inverse problem: For all possible assignments of T and F to UNION, INTER, COMP, find (if it is possible) a class of sets that is closed exactly under those that are assigned T. I would like the sets to be natural. I would also like to have some from Complexity theory, which I denote CT (e.g., Reg, P, R.E.) and some not (e.g., set of all infinite sets) which I denote HS for High School Student could come up with it. If I want other examples I will say OTHERS? We assume our universe is {a,b}^*.

1. UNION-YES, INTER-YES, COMP-YES:
   
   1. CT: Reg, P, Poly Hier, PSPACE, Prim Rec, Decidable, Arithmetic Hier.
      
   2. HS: Set of all subsets of {a,b}^*. OTHERS?
      
2. UNION-YES, INTER-YES, COMP-NO:
   
   1. CT: NP (probably), R.E. OTHERS?
      
   2. HS: Set of all finite subsets of {a,b}^*. OTHERS?
      
3. UNION-YES, INTER-NO, COMP-YES: NOT POSSIBLE.
   
4. UNION-YES, INTER-NO, COMP-NO.
   
   1. CT: Context Free Langs. OTHERS?
      
   2. HS: Set of all infinite subsets of {a,b}^*. OTHERS?
      
5. UNION-NO, INTER-YES, COMP-YES: NOT POSSIBLE.
   
6. UNION-NO, INTER-NO, COMP-YES:
   
   1. CT: The set of all regular langs that are accepted by a DFA with ≤ 2 states. (Can replace 2 with any n ≥ 2.)
      OTHERS?
      
   2. HS: Set of all subsets of {a,b}^* that are infinite AND their complements are infinite. OTHERS?
      
7. UNION-NO, INTER-YES, COMP-NO:
   
   1. CT: I can't think of any- OTHERS?
      
   2. HS: { emptyset, {a}, {b} }
      
8. UNION-NO, INTER-NO, COMP-NO:
   
   1. CT: The set of all reg language accepted by a DFA with ≤ 3 states and ≤ 2 accepting states. (Can replace (3,2) with other pairs) OTHERS?
      
   2. HS: I can't think if any- OTHERS?