On continuous polynomials of the Macías space
Jhixon Macías
TL;DR
The paper studies continuity of polynomial maps on the Macías space $M(\mathbb{N})$, a topology generated by the sets $\sigma_n=\{m\;|\;\gcd(n,m)=1\}$, contrasting with Golomb topology. The authors introduce a necessary and sufficient condition for continuity in terms of the prime-derived sets via $f^{-1}(\sigma_p)$, leveraging Dirichlet's theorem to demonstrate non-continuity for most multi-term polynomials. They prove the main result: the only continuous polynomials are the constant polynomials and monomials of the form $f(x)=a x^n$ with $a,n\in\mathbb{N}$; any polynomial with at least two nonzero terms and $a_0=0$ is discontinuous. The work clarifies how arithmetic structure interacts with the Macías topology and suggests extensions to function classes with modular consistency, highlighting that only certain exponential forms $a^x$ remain continuous in this setting. These findings contribute to a deeper understanding of continuity in arithmetic topologies and pave the way for further exploration of continuous maps on Golomb–Macías spaces.
Abstract
Let $\mathbb{N}$ be the set of natural numbers. The Macías space $M(\mathbb{N})$ is the topological space $(\mathbb{N},τ_M)$ where $τ_M$ is generated by the collection of sets $σ_n := \{ m \in \mathbb{N} : \gcd(n, m) = 1 \}$. In this paper, we characterize the continuity of polynomials over $ M(\mathbb{N})$ and prove that the only continuous polynomials are monomials