Abstract
A function f : X -> X determines a topology P(f) = {U subset of & nbsp; X : f(-1)(U) subset of & nbsp; U}. A topological space (X, tau) is primal (or functional Alexandroff ) if tau = P(f) for some function f, and is k -primal if tau is the supremum of a set of k primal topologies on X. Using the associated specialization quasiorder, we give necessary and sufficient conditions for a finite topological space to be k-primal. We show that the k-primal topologies on a finite set X form a lattice and discuss lattice complements.(c) 2021 Elsevier B.V. All rights reserved.