Some Properties of Primal Topologies Seen as Semirings - Garcia-Mendoza, Carlos; Vielma, Jorge; Játem, José (Topology Proceedings, 2023)
Given a set $X$ and a function $f: X → X$, a topology $τ_f$ is determined by taking the open sets to be those sets $A \subset X$ such that $f^{-1}(A) \subseteq A$. The topological space ( $X, τ_f$ ) is called primal space or functional Alexandroff space. In this paper, we study some properties of primal topologies seen as semirings. We prove, for example, that if ( $X, τ_f$ ) is a connected primal space, then $τ_f$ is a local semiring. We also determine some topological conditions for a square matrix A to be invertible, considering the primal topology $τ_A$ generated by the matrix.