Article

Linear topologies and sequential compactness in topological modules

Published in: Quaestiones Mathematicae
Volume 40, issue 7, 2017 , pages: 897–908
DOI: 10.2989/16073606.2017.1334716
Author(s): Francisco Javier García-PachecoDepartment of Mathematics, Spain, Pablo PiniellaDepartment of Mathematics, Spain

Abstract

We prove that an absolute semi-valued ring is first-countable if the set of invertibles is separable and its closure contains 0. We also show that every linearly topologized topological module over an absolute semi-valued ring whose invertibles approach 0 has the trivial topology. We also show that every sequentially compact set in a topological module is bounded if the module is over an absolute semi-valued ring whose set of invertibles is separable and its closure contains 0. Finally, we find sufficient conditions for a sequentially compact neighborhood of 0 to force the corresponding module to be finitely generated.

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