Is Fürstenberg's topology useful? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:53:00Z http://mathoverflow.net/feeds/question/42589 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42589/is-furstenbergs-topology-useful Is Fürstenberg's topology useful? I. J. Kennedy 2010-10-18T06:32:29Z 2010-10-19T11:54:22Z <p>It's hard not to be amused and perhaps even amazed when first encountering Fürstenberg's clever "topological" <a href="http://en.wikipedia.org/wiki/Furstenberg%27s_proof_of_the_infinitude_of_primes" rel="nofollow">proof</a> that there are infinitely many primes. Closer inspection, however, reveals the disappointing truth that there really isn't anything topological going on there, as pointed out by BCnrd in a comment to <a href="http://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts/42517#42517" rel="nofollow">this answer</a>.</p> <p>Nevertheless, the topology on $\mathbb{Z}$ introduced in the proof, where an open set is defined as any union of arithmetic sequences, does seem both natural and interesting.</p> <p>My question is this: Can anything useful be done with this topology? Useful would include a new theorem, a simplification to a proof of a known result, or even fresh insight into standard material.</p> http://mathoverflow.net/questions/42589/is-furstenbergs-topology-useful/42593#42593 Answer by S. Carnahan for Is Fürstenberg's topology useful? S. Carnahan 2010-10-18T07:59:49Z 2010-10-18T07:59:49Z <p>The answer to your question is yes, but it is a stretch to claim that the topology is due to Furstenburg. There is an extended discussion on Furstenburg's proof in the comments to <a href="http://mathoverflow.net/questions/34699/approaches-to-riemann-hypothesis-using-methods-outside-number-theory/34718#34718" rel="nofollow">this answer</a>. The short version is as Chandan Singh Dalawat said in the comments above: this topology on the integers is the profinite topology, and people had been studying profinite topologies long before Furstenburg.</p> <p>The topology is useful in the sense that profinite completions are useful. In particular, you may argue that it is a natural topology on the fundamental group of a circle (or the punctured complex affine line), since its profinite completion is the geometric fundamental group of the multiplicative group <code>$\mathbb{G}_m$</code>. It also appears in some form whenever one uses the ring of adeles <code>$\mathbb{A}_\mathbb{Q}$</code>, which you may encounter when studying Tate's thesis or automorphic representations.</p> http://mathoverflow.net/questions/42589/is-furstenbergs-topology-useful/42764#42764 Answer by Greg Kuperberg for Is Fürstenberg's topology useful? Greg Kuperberg 2010-10-19T11:15:42Z 2010-10-19T11:15:42Z <p>A more rigorous version of Scott's answer: If a topology on a group $G$ is translation-invariant, then it also defines a <em>uniformity</em> on $G$, by definition a distinguished set of neighborhoods of the diagonal $G \times G$ that is analogous to a metric. Actually, in the present example with $G = \mathbb{Z}$, the uniformity comes from a metric. Like the metric spaces that they generalize, uniform spaces have completions. The completion of $\mathbb{Z}$ with respect to the uniformity cited by Furstenberg is exactly the adelic profinite completion of $\mathbb{Z}$. Or if $G$ is any group, there is a similar topology generated by finite-index subgroups, and a uniformity, and the completion is the profinite completion.</p>