On an exotic topology of the integers

TL;DR

This paper explores the properties of Furstenberg's topology on integers, demonstrating its metrizability, total disconnectedness, and the topological ring structure, with applications to separating prime sets via arithmetic progressions.

Contribution

It establishes new topological properties of Furstenberg's topology and applies them to prime set separation, advancing understanding of integer topologies.

Findings

Furstenberg's topology is metrizable and totally disconnected.

The integers form a topological ring under this topology.

Disjoint prime sets can be separated by arithmetic progressions.

Abstract

We study some interesting properties of Furstenberg's topology of the integers. We show that it is metrizable, totally disconnected, and (Z,+,.) is a topological ring with respect to this topology. As an application, we show that any two disjoint sets of primes can be separated by arithmetic progressions.