On Wikipedia, there is a list of properties of sets of reals, which are in some sense "interesting": just have a look.

I could not find a comparable list of properties of sets of natural numbers (except their classification in the arithmetical hierarchy).

a) Any suggestions where to find such a list? (Arithmetic analogon to "Descriptive Set Theory"?)

b) Any reasons why (properties of) sets of natural numbers are less "interesting" than (properties of) sets of reals? Is it simply because the basic property of "open-ness" is missing for sets of natural numbers?