Top(X) y Spec(τ ) como espacios primales

• Viviana Benavides Instituto de Posgrado, Universidad Técnica de Manabı́, Portoviejo, Manabı́, Ecuador https://orcid.org/0000-0002-4984-4613
• Jorge Enrique Vielma Departamento de Matemáticas, Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral Guayaquil, Guayas, Ecuador https://orcid.org/0000-0001-9620-6756

Keywords: Primal topology, prime spectrum, semiring, Alexandroff topology

Abstract

An Alexandroff topology can be defined over a non-empty set \(X\), through a function \(f:X\to X\), deciding that the open sets are those subsets \(A\subseteq X\) that contain their preimage, that is \(\tau_f:=\{A\subset X: f^{-1}(A)\subseteq A\}\). This topology is called primal topology and the space \((X, \tau_f)\) is called primal space. In this work we explore a primal topology \(\tau_\psi\) induced on \(Top(X)\), through the funtion \(\psi: Top(X)\to Top(X)\) defined as \(\psi(\tau)=\overline{\tau}\), with \(\overline{\tau}\) the closure of \(\tau\) in \(2^X\) with the product topology. It is shown that the set of all Alexandroff topologies in \(Top(X)\) is dense in \((Top(X),\tau_{\psi^*})\), with \(\tau_{\psi^*}\) the cotopology. It is also shown that the set \(\phi(\tau):=\{A\in \tau_\psi :\tau\notin A\}\) is a maximal ideal of \(\tau_\psi\) if and only if \(\tau\) is Alexandroff. Finally we explore the primal topologies in the prime spectrum of a semiring.

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