It is well-known that

A: The series of the reciprocals of the primes diverges

My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers.

Property A tells us that the primes are a rather fat subset of $\mathbb{N}$. Is there a way to define a topology on $\mathbb{N}$ such that every dense subset of $\mathbb{N}$ (under this topology) corresponds to a fat subset of the natural numbers?

What do you think about this?

Yes, it's possible.  Define the closed sets to be the sets the sum of whose reciprocals converges, together with $\mathbb{N}$.  This collection of subsets is closed under arbitrary intersection and finite union, so it does form the closed sets of a topology.  

A subset of $\mathbb{N}$ is dense in this topology if its closure is $\mathbb{N}$, in other words, if it is not contained in any smaller closed set -- in other words, if it is not contained in any set the sum of whose reciprocals converges.  This is equivalent to the sum of its reciprocals not converging.   
