Let's define this topology in $\Bbb N$ (here $\Bbb N$ begins at $1$):
$$K\subset\Bbb N\text{ is closed }\iff K=\Bbb N\;\text{ or }\;\sum_{n\in K}\frac1n\text{ converges}$$
I have worked some on it. For example, it is not Hausdorff. I'm not sure whether it is compact or not.
My question is: is there some known relevant results about this topology? Feel free to edit tags according to your answers.