Filters and limits in filtered spaces

Abstract

In this lecture  notes,  we study filters and limits in a feasible  sufficiently general  way for  many  applications  to  Physics,  Economics and  Social  Science.    In particular  we analyze: Filters; Filter bases; Basic example in topological  spaces; Basic examples in preordered spaces; Filtered spaces; Induced filter bases; Examples in topological spaces; Examples on the real line; Limits; Limits with respect to filter bases; Limits with respect to filter bases  on subspaces; Limits and topology; Limits of functions on topological  spaces; Limits of functions on a topological  subspace; Some basic theorems for topology; Unicity of the limit in Hausdor spaces; Limits and functionals; Limits for real functional; Limits and operations; Local boundness; Complements; Limits and convergence of filtered bases; Limit of composition.