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On continuous self-maps and homeomorphisms of the Golomb space. (English) Zbl 06997360
Summary: The Golomb space \(\mathbb{N}_\tau\) is the set \(\mathbb{N}\) of positive integers endowed with the topology \(\tau\) generated by the base consisting of arithmetic progressions \(\{a+bn:n\geq 0\}\) with coprime \(a,b\). We prove that the Golomb space \(\mathbb{N}_\tau\) has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set \(\Pi\) of prime numbers is a dense metrizable subspace of \(\mathbb{N}_\tau\), and each homeomorphism \(h\) of \(\mathbb{N}_\tau\) has the following properties: \(h(1)=1\), \(h(\Pi)=\Pi\), \(\Pi_{h(x)}=h(\Pi_x)\), and \(h(x^{\mathbb{N}})=h(x)^{\mathbb{N}}\) for all \(x\in\mathbb{N}\). Here \(x^{\mathbb{N}}:=\{x^n\colon n\in\mathbb{N}\}\) and \(\Pi_x\) denotes the set of prime divisors of \(x\).

MSC:
54D05 Connected and locally connected spaces (general aspects)
11A41 Primes
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