A series of series topologies on $\mathbb{N}$

Abstract

Each series $\sum_{n=1}^\infty a_n$ of real positive terms gives rise to a topology on $\mathbb{N} = \{1,2,3,...\}$ by declaring a proper subset $A\subseteq \mathbb{N}$ to be closed if $\sum_{n\in A} a_n < \infty$. We explore the relationship between analytic properties of the series and topological properties on $\mathbb{N}$. In particular, we show that, up to homeomorphism, $|\mathbb{R}|$-many topologies are generated. We also find an uncountable family of examples $\{\mathbb{N}_\alpha\}_{\alpha \in [0,1]}$ with the property that for any $\alpha < \beta$, there is a continuous bijection $\mathbb{N}_\beta\rightarrow \mathbb{N}_\alpha$, but the only continuous functions $\mathbb{N}_\alpha\rightarrow \mathbb{N}_\beta$ are constant.