Primal topologies on the integers

Abstract

Given an innite set X and a function f : X→ X, the primal topology on X induced by f is the topology T_f = {U C X : f ^-1(U)C U}. In this paper, we
prove that there are 2^w pairwise non-homemomorphic primal topologies ℕ. We also prove that an infnite set cannot have more than 2^w 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.

Key words: Primal topologies, orbits, periodic points prime orbits.