O’Brien, M.Troitsky, V.G.Van der Walt, Jan Harm2023-01-262023-01-262023M. O’Brien, V.G. Troitsky & J.H. van der Walt (2023): Net convergence structures with applications to vector lattices, Quaestiones Mathematicae, 46(2): 243–280, DOI: 10.2989/16073606.2021.2012721.1607-3606 (print)1727-933X (online)10.2989/16073606.2021.2012721https://repository.up.ac.za/handle/2263/88964Convergence is a fundamental topic in analysis that is most commonly modeled 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.en© 2022 NISC (Pty) Ltd. This is an electronic version of an article published in Quaestiones Mathematicae, vol. 46, no. 2, pp. 243-280, 2023. doi : 10.2989/16073606.2021.2012721. Quaestiones Mathematicae is available online at: https://www.tandfonline.com/loi/tqma20.Convergence structuresNets and filtersVector latticesPreprint Article