AI 中文总结
本文研究非零高斯整数集上的互素拓扑，刻画其拓扑不可区分性、连通性与Kolmogorov商结构，并通过高斯素数的稠密性给出其无穷性的拓扑证明。
AI 中文摘要
本文研究非零高斯整数集$\mathbb{Z}[i]\setminus\{0\}$上的互素拓扑，该拓扑由算术基$σ_α=\{β\neq 0 : \gcd(α,β)\sim 1\}$生成。通过分析至多相伴的素支撑，我们证明两个点拓扑不可区分当且仅当它们的支撑重合。我们证明该空间既是超连通的也是超连通的（hyperconnected且ultraconnected），并明确将其Kolmogorov商$X$刻画为高斯素数相伴类的有限子集构成的空间，赋予基$\mathcal{O}_F=\{S\in X:S\cap F=\emptyset\}$，其中$F\in X$。最后，我们证明高斯素数集合在互素拓扑下在$\mathbb{Z}[i]\setminus\{0\}$中稠密，由此给出高斯素数无穷性的拓扑证明。
英文摘要
This article investigates the coprimality topology on the set of non-zero Gaussian integers, $\mathbb{Z}[i]\setminus\{0\}$, generated by the arithmetic basis $σ_α=\{β\neq 0 : \gcd(α,β)\sim 1\}$. 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.
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