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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.

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Hint:

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

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