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

TL;DR

This paper investigates topologies on the natural numbers generated by series of positive real numbers, revealing a rich structure with many non-homeomorphic topologies and interesting continuous function properties.

Contribution

It introduces a family of topologies on  generated by series and analyzes their topological and analytic relationships, including the classification up to homeomorphism.

Findings

Uncountably many topologies are generated by different series.

Existence of a family with continuous bijections in one direction but only constant functions in the other.

Topologies are classified up to homeomorphism with cardinality of the continuum.

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.