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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.