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The Golomb space is topologically rigid

The $Golomb$ $space$ $\mathbb N_τ$ is the set $\mathbb N$ of positive integers endowed with the topology $τ$ generated by the base consisting of arithmetic progressions $\{a+bn:n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space $\mathbb N_τ$ is topologically rigid in the sense that its homeomorphism group is trivial. This resolves a problem posed by the first author at Mathoverflow in 2017.