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 n=1an of real strictly positive terms gives rise to a topology on = {1,2,3,} by declaring a proper subset A to be closed if nAan < . We explore the relationship between analytic properties of the series and topological properties on . In particular, we show that, up to homeomorphism, ||-many topologies are generated. We also find an uncountable family of examples {α}α[0,1] with the property that for any α < β, there is a continuous bijection β α, but the only continuous functions α β are constant.

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