Abstract:
Convergence 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.
