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One of the strange proofs (among the other beautiful proof) in the book "Proofs from the book" is the fifth one, which uses a special topology on the set of integer numbers, to prove there are infinite prime numbers.

My question is:

Is this method special just for this case, or is there anything deeper and this technique (or its generalization) can be used for some other kinds of problems?

A related question (by @ToddTrimble comment):

Is Fürstenberg's topology useful?

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    $\begingroup$ When unwound straightforwardly, the "topological" proof is really just the usual proof in disguise (and it doesn't use anything more than the language of topology in the first place). That said, I am under the impression that the topology introduced there is actually interesting, just not really for that reason. $\endgroup$
    – Noah Schweber
    Commented Mar 12, 2020 at 15:27
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    $\begingroup$ If anybody could mention which topology it is, it would help answering the question. (If it's the profinite one, it definitely has a huge number of uses, say $p$-adic numbers and so on.) $\endgroup$
    – YCor
    Commented Mar 12, 2020 at 15:55
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    $\begingroup$ Certainly it's the profinite topology, i.e., the topology inherited from the profinite completion $\hat{\mathbb{Z}}$ that appears in the adeles, as pointed out by Chandan Singh Dalawat here: mathoverflow.net/q/42589/2926 $\endgroup$
    – Todd Trimble
    Commented Mar 12, 2020 at 16:00
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    $\begingroup$ It's your call. I'm not suggesting you should. Maybe someone would like to say something even more enlightening than the hints given so far in comments; dunno. $\endgroup$
    – Todd Trimble
    Commented Mar 12, 2020 at 16:12
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    $\begingroup$ PS the "Furstenberg topology" is the terminology given by some subcommunity to the profinite topology. $\endgroup$
    – YCor
    Commented Mar 12, 2020 at 16:59

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