We investigate properties of coset topologies on commutative domains with an identity, in particular, the S-coprime topologies de ned by Marko and Poru-bsky (2012) and akin to the topology de ned by Furstenberg (1955) in his proof of the in nitude of rational primes. We extend results about the in nitude of prime or maximal ideals related to the Dirichlet theorem on the in nitude of primes from Knopfmacher and Porubsky (1997), and correct some results from that paper. Then we determine cluster points for the set of primes and sets of primes appearing in arithmetic progressions in S-coprime topologies on ℤ. Finally, we give a new proof for the infinitude of prime ideals in number fields.
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