Locally hulled topologies

TL;DR

This paper establishes a general method for generating group topologies using pseudo-norms, applicable to various structures like vector spaces, lattices, and Boolean algebras, based on certain closed-base conditions.

Contribution

It introduces a unifying framework for constructing group topologies via pseudo-norms from closed-base topologies, encompassing multiple classical topologies.

Findings

Applicable to linear and locally convex topologies on vector spaces

Includes locally solid and Fatou topologies on vector lattices

Covers Fréchet-Nikodým topologies on Boolean algebras

Abstract

We present a general result about generating group topologies by pseudo-norms. Namely, we show that if a topology has a base of sets which are closed in a certain sense, then it can be generated by a collection of pseudo-norms such that the balls in these pseudo-norms are also closed in the same sense. The examples include linear and locally convex topologies on vector spaces, locally solid and Fatou topologies on vector lattices and Fr\'echet-Nikod\'ym topologies on Boolean algebras.