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

Abstract

Each series ${\sum }_{n=1}^{\infty }{a}_n$ of real strictly positive terms gives rise to a topology on $\mathbb{N}=\left\{1,2,3,\dots \right\}$ 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, $\left|ℝ\right|$-many topologies are generated. We also find an uncountable family of examples ${\left\{{\mathbb{N}}_{\alpha }\right\}}_{\alpha \in \left[0,1\right]}$ with the property that for any $\alpha <\beta$, there is a continuous bijection ${\mathbb{N}}_{\beta }\to {\mathbb{N}}_{\alpha }$, but the only continuous functions ${\mathbb{N}}_{\alpha }\to {\mathbb{N}}_{\beta }$ are constant.