Primal topologies on the integers

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 n...

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Bibliographic Details
Published in:Quaestiones mathematicae 2021-04, Vol.44 (4), p.435-445
Main Authors: García-Ferreira, S., Guale, A., Vielma, J.
Format: Article
Language:English
Subjects:
orbits
periodic points prime orbits
Primal topologies
Primary: 54C99
Topology
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Summary: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.
ISSN:1607-3606
1727-933X
DOI:10.2989/16073606.2019.1695686