# $\forall A\subset \mathbb{N}$ the sum of the reciprocals of $A$ diverges iff $A$ is $(\tau, \mathbb{N})$-dense

Show that is possible to endow the natural numbers with a topology $\tau$ such that for every $A\subset \mathbb{N}$ the sum of the reciprocals of $A$ diverges iff $A$ is $(\tau, \mathbb{N})$-dense.

A nonempty subset $U$ of $\Bbb N$ is open iff $$\sum_{n\notin U}\frac1n<\infty$$