The emergence of the concept of filter in topological categories

Abstract

In all approaches to convergence where the concept of filter is taken as primary, the usual motivation is the notion of neighborhood filter in a topological space. However, these approaches often lead to spaces more general than topological ones, thereby calling into question the need to use filters in the first place. In this note we overturn the usual view and take as primary the notion of convergence in the most general context of centered spaces. In this setting, the notion of filterbase emerges from the concept of germ of a function, while the concept of filter emerges from an amnestic modification of the subcategory of centered spaces admitting germs at each point.