Vol. 13, No. 2, 2020

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

Jason DeVito and Zachary Parker

Vol. 13 (2020), No. 2, 205–218
Abstract

Each series ${\sum }_{n=1}^{\infty }{a}_{n}$ of real strictly positive terms gives rise to a topology on $ℕ=\left\{1,2,3,\dots \phantom{\rule{0.3em}{0ex}}\right\}$ 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 ${\left\{{ℕ}_{\alpha }\right\}}_{\alpha \in \left[0,1\right]}$ with the property that for any $\alpha <\beta$, there is a continuous bijection ${ℕ}_{\beta }\to {ℕ}_{\alpha }$, but the only continuous functions ${ℕ}_{\alpha }\to {ℕ}_{\beta }$ are constant.

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Keywords
series, countable topologies
Mathematical Subject Classification 2010
Primary: 54A10, 54G99