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Mackey hyperplanes/enlargements for Tweddle’s space

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

Tweddle (India J Math 50:99–114, 2008) showed that Banach’s bounded sequence space retains certain properties when endowed with a topology \(\eta \) defined by a maximal almost disjoint (mad) family of natural numbers. In particular, it is Mackey. But it lacks most weak barrelledness properties, forcing Saxon and Tweddle (Adv Math 145:230–238, 1999) to find elsewhere the first example of a Mackey \(\aleph _{0}\)-barrelled space that is not barrelled. We shall show that \(\eta \) varies in tightness as the size of its defining mad family: Tweddle’s spaces are not all isomorphic. But their countable-codimensional subspaces and countable enlargements are all Mackey. Likewise for the Saxon and Tweddle example, which multiplies our scant supply of such. Independently interesting methods are developed.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, P.O. Box 118105, Gainesville, FL, 32611-8105, USA

    Stephen A. Saxon

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  1. Stephen A. Saxon
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Correspondence to Stephen A. Saxon.

Additional information

For inspiration and support, thanks to Ian Tweddle, Aaron Todd, The New York Seminar on General Topology and Topological Algebra, and Baruch College. Thanks to Jerzy Kąkol and Luis Sánchez Ruiz for correcting the attribution of Lemma 1.

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Saxon, S.A. Mackey hyperplanes/enlargements for Tweddle’s space. RACSAM 108, 1035–1054 (2014). https://doi.org/10.1007/s13398-013-0159-x

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