# Reference request: A collection of topologies on $\mathbb{N}$ formed via series

First, some quick notation: for any series $\sum_{n=1}^\infty a_n$ whose terms are positive real numbers, and for any subset $M = \{m_1, m_2,...\} \subseteq \mathbb{N}$, we write $\sum_M a_n$ to mean $a_{m_1} + a_{m_2} + ....$.

Then, for each series, one can associate a topology on $\mathbb{N}$ by declaring a proper subset $M\subsetneq \mathbb{N}$ to be closed iff $\sum_M a_n < \infty$. So, e.g., in the $\sum \frac{1}{n}$-topology, the set of even numbers is neither open nor closed (both $\sum \frac{1}{2n+1}$ and $\sum \frac{1}{2n}$ diverge), but the set of squares is closed ($\sum \frac{1}{n^2}$ converges).

There is, of course, an interplay between the topological properties $\mathbb{N}$ inherits and the analytic properties of $\sum a_n$. For instance, $\mathbb{N}$ gets the cofinite topology iff $\liminf a_n > 0$.

Much less trivially, one can show that this association generates precisely $|\mathbb{R}|$ many topologies on $\mathbb{N}$ (distinct up to homeomorphism). In fact, for $0\leq p < q \leq 1$, the spaces obtained from $\sum \frac{1}{x^p}$ and $\sum \frac{1}{x^q}$ are not homeomorphic.

I am not a general topologist by training so my question is probably very simple. Has anything like this appeared in the literature before? MathSciNet doesn't seem to turn up anything, but maybe there is some terminology I need to know. Maybe these form a subclass in some well studied class of topological spaces?

In case it helps, here are some curious data points about these topological spaces.

1. They are homogeneous. More generally, the homeomorphism group acts transitively on the set of $k$-element subsets for any fixed $k$.
2. Every subset is closed or dense. (This is equivalent to the topology being downward-closed: every subset of a proper closed set is closed.)
3. If $\lim_{n\rightarrow \infty} a_n = 0$ but $\sum a_n$ diverges, then any non-empty open set is homeomorphic to $\mathbb{N}$.
• What you've described as a topology, is actually a family of analytic P-ideals on $P(\mathbb{N})$ called the "summable ideals." They are studied for different reasons, and the general search terms to use are "summable ideal" or more generally "analytic P-ideal". – Not Mike Jul 17 '18 at 23:05
• @Not Mike: Wow - some quick googling has turned up a ton of really relevant results. Time to dust of my set theory! Care to turn your comment to an answer? – Jason DeVito Jul 17 '18 at 23:41
• Will do. Thanks. And yes, there are a lot of nice results dealing with P-ideals. – Not Mike Jul 17 '18 at 23:45
• @JasonDeVito Is the statement here true ? – Duchamp Gérard H. E. Jul 18 '18 at 7:20
• @DuchampGérardH.E. I don't know - I first saw that question last night after googling Not Mike's suggestion! I will think about ti a bit, but my measure theory is really weak. (I'm a differential geometer by trade!) – Jason DeVito Jul 18 '18 at 12:38

What you've described as a topology, is actually a family of analytic P-ideals on $P(\mathbb{N})$ called the "summable ideals." They are studied for different reasons, and the general search terms to use are "summable ideal" or more generally "analytic P-ideal".