The emergence of the concept of filter in topological categories

TL;DR

This paper explores the foundational role of filters in topological categories by examining convergence in centered spaces, revealing how filters naturally arise from germs of functions and their modifications.

Contribution

It redefines the emergence of filters as fundamental in convergence theory by analyzing centered spaces and germ concepts, challenging traditional topological perspectives.

Findings

Filters originate from germs of functions in centered spaces

The concept of filter emerges from an amnestic modification of centered spaces

Provides a new foundational perspective on convergence in topology

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.