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We provide a construction which covers as special cases many of thetopologies on integers one can find in the literature. Moreover, our analysisof the Golomb and Kirch topologies inserts them in a family of connected,Hausdorff topologies on $\mathbb{Z}$, obtained from closed sets of theprofinite completion $\hat{\mathbb{Z}}$. We also discuss various applicationsto number theory.

Coset topologies on $\mathbb{Z}$ and arithmetic applications