Primal topologies on the integers

Abstract

Given an infinite set X and a function f : X → X, the primal topology on X induced by f is the topology τ _f = {U ⊆ X : f ⁻¹(U ) ⊆ U}. In this paper, we prove that there are 2 ^ω pairwise non-homemomorphic primal topologies on ℕ. We also prove that an infinite set cannot have more than 2 ^ω pairwise non-homeomorphic primal topologies. We give a necessary and sufficient condition to guarantee that an Alexandroff topology be n-resolvable for an 2 ≤ n ∈ ℕ. Other results on primal topologies are also given.