arXiv subjects
Publications and source records attributed to Jhixon Macías.
In this paper, we study some properties of the closure operator in the Macías topology on infinite integral domains. Moreover, under certain conditions, we present topological proofs of the infiniteness of maximal ideals and non-associated irreducible elements, taking advantage of the hyperconnectedness of the Macías topology. Additionally, some problems are proposed.
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
In this paper, we study some properties of self-homeomorphisms on the Macías topology over $\mathbb{N}$, and we demonstrate that this space is not topologically rigid.
In this manuscript a recent topology on the positive integers generated by the collection of $\{σ_n:n\in\mathbb{N}\}$ where $σ_n:=\{m: \gcd(n,m)=1\}$ is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.
We present a new topological proof of the infinitude of prime numbers with a new topology. Furthermore, in this topology, we characterize the infinitude of any non-empty subset of prime numbers.
Let $p$ be a prime number, and $h$ a positive integer such that $\gcd(p,h)=1$. We prove, without invoking Dirichlet's theorem, that the arithmetic progression $p\left(\mathbf{N}\cup \{0\}\right)+h$ contains infinitely many prime numbers. This is a special case of Dirichlet's theorem not considered by other authors.