On continuous polynomials of the Mac\'ias space

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