A series of series topologies on ℕ

DeVito, Jason; Parker, Zachary

doi:10.2140/involve.2020.13.205

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

Jason DeVito and Zachary Parker

Vol. 13 (2020), No. 2, 205–218

DOI: 10.2140/involve.2020.13.205

Abstract

Each series $\sum_{n = 1}^{\infty} a_{n}$ of real strictly positive terms gives rise to a topology on $ℕ = {1, 2, 3, \dots}$ by declaring a proper subset $A \subseteq ℕ$ 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 $ℕ$ . In particular, we show that, up to homeomorphism, $| ℝ |$ -many topologies are generated. We also find an uncountable family of examples ${ℕ_{α}}_{α \in [0, 1]}$ with the property that for any $α < β$ , there is a continuous bijection $ℕ_{β} \to ℕ_{α}$ , but the only continuous functions $ℕ_{α} \to ℕ_{β}$ are constant.

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Keywords

series, countable topologies

Mathematical Subject Classification 2010

Primary: 54A10, 54G99

Milestones

Received: 27 February 2019

Revised: 7 July 2019

Accepted: 3 December 2019

Published: 30 March 2020

Communicated by Józef H. Przytycki

Authors

Jason DeVito
University of Tennessee at Martin Martin, TN United States

Zachary Parker
University of Tennessee at Martin Martin, TN United States

Vol. 13, No. 2, 2020