Primal Topologies on Finite‐Dimensional Vector Spaces Induced by Matrices
Luis Fernando Mejías, Jorge Enrique Vielma, Ángel Guale, Ebner Pineda · International Journal of Mathematics and Mathematical Sciences · 2023
Given an matrix A, considered as a linear map A : ℝn⟶ℝn, then A induces a topological space structure on ℝn which differs quite a lot from the usual one (induced by the Euclidean metric). This new topological structure on ℝn has very interesting properties with a nice special geometric flavor, and it is a particular case of the so called “primal space,” In particular, some algebraic information can be shown in a topological fashion and the other way around. If X is a non‐empty set and f : X⟶X is a map, there exists a topology τf induced on X by f, defined by τf = {U ⊂ X : f−1(U) ⊂ U}. The pair (X, τf) is called the primal space induced by f. In this paper, we investigate some characteristics of primal space structure induced on the vector space ℝn by matrices; in particular, we describe geometrical properties of the respective spaces for the case.