Is there a non-trivial topological group structure of \$\mathbb{Z}\$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:23:52Z http://mathoverflow.net/feeds/question/18433 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18433/is-there-a-non-trivial-topological-group-structure-of-mathbbz Is there a non-trivial topological group structure of \$\mathbb{Z}\$? Cristos A. Ruiz 2010-03-16T22:19:53Z 2010-03-16T22:39:00Z <p>More specificaly, is there a haussdorf non-discrete topology on \$\mathbb{Z}\$ that makes it a topological group with the usual addition operation?</p> http://mathoverflow.net/questions/18433/is-there-a-non-trivial-topological-group-structure-of-mathbbz/18435#18435 Answer by James for Is there a non-trivial topological group structure of \$\mathbb{Z}\$? James 2010-03-16T22:25:57Z 2010-03-16T22:25:57Z <p>Yes. Take, for example, the subgroups \$p^k\mathbb{Z}\$, for \$k>0\$ and a fixed prime \$p\$, as a basis of neighborhoods of the identity.</p> http://mathoverflow.net/questions/18433/is-there-a-non-trivial-topological-group-structure-of-mathbbz/18437#18437 Answer by Mariano Suárez-Alvarez for Is there a non-trivial topological group structure of \$\mathbb{Z}\$? Mariano Suárez-Alvarez 2010-03-16T22:39:00Z 2010-03-16T22:39:00Z <p>There is a topology on \$\mathbb Z\$ which has the set of all arithmetic sequences as a basis. It shows up in the topological proof of the infinitude of primes, <em>cf.</em> [H. Fürstenberg, On the Infinitude of Primes, Amer. Math. Monthly 62 (1955), 353]</p>