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The pseudometric topology induced by upper asymptotic density

Jonathan M. Keith

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).

The pseudometric topology induced by upper asymptotic density

Abstract

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

Paper Structure

This paper contains 4 sections, 5 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Completeness of $({\mathcal{D}}, d)$
  3. Closed subsets of ${\mathcal{D}}$
  4. Applications

Key Result

Theorem 1.1

$({\mathcal{P}}({\mathbb{N}}), d)$ is complete and ${\mathcal{D}}$ is closed.

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof