Convergence spaces: nets vs filters

Every topological space has a natural concept of convergence. However, there are important convergences that are not generated by a topology, e.g., convergence almost everywhere. In particular, non-topological convergences are commonly used in the theory of vector lattices. This issue was resolved in the 50s by creating a theory of convergence spaces, which generalizes Topology. However, this theory was initially designed in the language of filters. This makes it less convenient for applications in Analysis, as many results in Analysis (in particular, in vector lattices) are usually stated in the language of sequences or nets (generalized sequences). In the talk, I will present a formulation of the theory of convergence spaces in the language of nets. On one hand, this theory is equivalent to the classical theory of filter convergence, so that all the body of results that have been developed in that theory remains applicable. On the other hand, the new theory allows for easier applications in Analysis.