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

Abstract

This article investigates the coprimality topology on the set of non-zero Gaussian integers, $\mathbb{Z}[i]\setminus\{0\}$, generated by the arithmetic basis $\sigma_\alpha=\{\beta\neq 0 : \gcd(\alpha,\beta)\sim 1\}$. By analyzing prime supports up to associates, we establish that two points are topologically indistinguishable if and only if their supports coincide. We demonstrate that the space is both hyperconnected and ultraconnected, and we explicitly characterize its Kolmogorov quotient $X$ as the space of finite subsets of associate classes of Gaussian primes, endowed with the basis $\mathcal{O}_F=\{S\in X:S\cap F=\emptyset\}$ for $F\in X$. Finally, we demonstrate that the set of Gaussian primes is dense in $\mathbb{Z}[i]\setminus\{0\}$ with respect to the coprimality topology, thereby establishing a topological proof of the infinitude of Gaussian primes.