The pseudometric topology induced by upper asymptotic density

TL;DR

This paper explores a pseudometric derived from upper asymptotic density on the power set of natural numbers, establishing its completeness and characterizing closed sets within this framework.

Contribution

It introduces a pseudometric based on upper asymptotic density, proves the completeness of the space, and characterizes closed subsets of sets with asymptotic density.

Findings

The pseudometric makes the power set of natural numbers complete.

The collection of sets with asymptotic density is closed under this pseudometric.

Closed subsets of these sets are characterized by a generalized additivity property.

Abstract

Upper asymptotic density induces a pseudometric on the power set of the natural numbers, with respect to which $P(\mathbb{N})$ is complete. The collection $D$ of sets with asymptotic density is closed in this pseudometric, and closed subsets of $D$ are characterised by a generalisation of an additivity property (AP0).