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arXiv · 2103.01339

Net convergence structures with applications to vector lattices

M. O'Brien·V. G. Troitsky·J. H. van der Walt
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Abstract

Convergence is a fundamental topic in analysis that is most commonly modelled using topology. However, there are many natural convergences that are not given by any topology; e.g., convergence almost everywhere of a sequence of measurable functions and order convergence of nets in vector lattices. The theory of convergence structures provides a framework for studying more general modes of convergence. It also has one particularly striking feature: it is formalized using the language of filters. This paper develops a general theory of convergence in terms of nets. We show that it is equivalent to the filter-based theory and present some translations between the two areas. In particular, we provide a characterization of pretopological convergence structures in terms of nets. We also use our results to unify certain topics in vector lattices with general convergence theory.

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M. O'Brien, V. G. Troitsky, J. H. van der Walt. 2021-03-01. Net convergence structures with applications to vector lattices. https://arxiv.org/abs/2103.01339

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