Filters and limits in filtered spaces

In this lecture notes, we study lters 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.