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?



On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...
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. 


