Is there a non-trivial topological group structure of $\mathbb{Z}$? - MathOverflow most recent 30 from http://mathoverflow.net2013-05-23T08:23:52Zhttp://mathoverflow.net/feeds/question/18433http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/18433/is-there-a-non-trivial-topological-group-structure-of-mathbbzIs there a non-trivial topological group structure of $\mathbb{Z}$?Cristos A. Ruiz2010-03-16T22:19:53Z2010-03-16T22:39:00Z
<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#18435Answer by James for Is there a non-trivial topological group structure of $\mathbb{Z}$?James2010-03-16T22:25:57Z2010-03-16T22:25:57Z<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#18437Answer by Mariano Suárez-Alvarez for Is there a non-trivial topological group structure of $\mathbb{Z}$?Mariano Suárez-Alvarez2010-03-16T22:39:00Z2010-03-16T22:39:00Z<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>