For instance, this is the case with defining $U \subset \Bbb{N}$ to be open iff $\sum_{x \notin U} \frac{1}{x} \lt \infty$ if we let $\sum_{x \notin \Bbb{N}} = 0$. I can't seem to get $\varnothing$ to be part of that "topology" as $\sum_{x \in \Bbb{N}} = \infty$. So what is this "pseudo-topology" called?
Please also see and comment on this post: $\forall A\subset \mathbb{N}$ the sum of the reciprocals of $A$ diverges iff $A$ is $(\tau, \mathbb{N})$-dense
In this case, the fact that the emptyset is not "open" is the only barrier to your definition yielding a topology. Closure under arbitrary nonempty unions is obvious; to see that the intersection of two "opens" is "open," let $S_X=\sum_{n\not\in X}{1\over n}$; then note that $$S_{X\cap Y}\le S_X+S_Y<\infty$$ if $S_X, S_Y<\infty$. In such situations as these, it's common to just add the emptyset so you get a genuine topology. Since this is really a trivial modification, I don't believe there is a name for someething which would be a topology if it contained $\emptyset$ (that is, for a family of subsets of some set $X$ which is closed under arbitrary nonempty unions, finite intersections, and contains $X$ itself).