Coprimality Topology on Gaussian Integers: Kolmogorov Quotient and Gaussian Prime Density

Overview

This article explores the coprimality topology defined on the set of non-zero Gaussian integers, denoted as $\mathbb{Z}[i]\setminus\{0\}$. The investigation focuses on the topological properties of this space, including hyperconnectedness, ultraconnectedness, and the characterization of its Kolmogorov quotient. A key finding involves demonstrating the density of Gaussian primes within this topological structure, which offers a topological proof for their infinitude.

Research Context

The study establishes a coprimality topology on $\mathbb{Z}[i]\setminus\{0\}$ using an arithmetic basis. This basis, denoted $\sigma_\alpha$, is defined as the set of non-zero Gaussian integers $\beta$ such that $\gcd(\alpha,\beta)\sim 1$. The analysis considers prime supports up to associates. The primary objective is to understand the topological relationships within this set, particularly how elements are related based on their coprimality properties.

Approach

The research employed a topological approach to analyze the set of non-zero Gaussian integers. The methodology involved defining a specific arithmetic basis, $\sigma_\alpha = \{\beta\neq 0 : \gcd(\alpha,\beta)\sim 1\}$, to generate the coprimality topology. The study then proceeded to analyze the topological indistinguishability of points by examining their prime supports, considering associates. This involved demonstrating specific properties of the space, such as hyperconnectedness and ultraconnectedness. A significant part of the approach was the explicit characterization of the Kolmogorov quotient. Finally, the methodology involved analyzing the distribution of Gaussian primes within this defined topology to establish their density.

Findings

• Two points within the coprimality topology are topologically indistinguishable if and only if their prime supports coincide, considering associates.
• The space endowed with the coprimality topology is characterized as both hyperconnected and ultraconnected.
• The Kolmogorov quotient $X$ of this space is explicitly characterized. It is identified as the space of finite subsets of associate classes of Gaussian primes. This quotient space is endowed with the basis $\mathcal{O}_F=\{S\in X:S\cap F=\emptyset\}$ for any finite subset $F$ of associate classes of Gaussian primes.
• The set of Gaussian primes is dense in $\mathbb{Z}[i]\setminus\{0\}$ with respect to the coprimality topology.
• The density of Gaussian primes in this topology provides a topological proof for the infinitude of Gaussian primes.