On continuous polynomials of the Macías space

Abstract

Let $\mathbb{N}$ be the set of natural numbers. The Macías space $M(\mathbb{N})$ is the topological space $(\mathbb{N},\ddot{I}{„}_M)$ where $\ddot{I}{„}_M$ is generated by the collection of sets $\ddot{I}{ƒ}_n:=\{m\in \mathbb{N}:\gcd (n,m)=1\}$. In this paper, we characterize the continuity of polynomials over $M(\mathbb{N})$ and prove that the only continuous polynomials are monomials