Timeline for Is there a connected $T_2$-topology on $\mathbb{Q}$ that is coarser than the Euclidean one?
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20 events
| when toggle format | what | by | license | comment | |
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| Jul 1, 2018 at 12:46 | comment | added | Taras Banakh | @PeteL.Clark Thank you for joining to the discussion. I completely agree that Solomon Golomb was a great (applied) mathematician but popularization of math also was his very strong and important preoccupation. | |
| Jul 1, 2018 at 8:25 | comment | added | Pete L. Clark | Let me save my vehemence for the following point: Solomon W. Golomb was most certainly a professional mathematician! (He was also an electrical engineer.) See en.wikipedia.org/wiki/Solomon_W._Golomb for some information about him, including: he got a PhD in mathematics from Harvard and was elected a fellow of the AMS. Moreover MathSciNet records 144 publications and 1041 citations. I will however venture to say, as someone who has spent some time looking at his work, that Golomb's ideas on "arithmetica topologica" turned out not to have the depth and significance that he hoped. | |
| Jul 1, 2018 at 8:17 | comment | added | Pete L. Clark | For whatever it is worth (very little, I believe), in the CLLP paper linked to in Banakh's above comment, we say "adic topology" rather than "Furstenberg topology." But the fact that this topology on $\mathbb{Z}$ is the one inherited from the profinite completion (or the ring of finite adeles) and thus predates Furstenberg is in my opinion not a major point, as what Chevalley and others were doing with adeles was quite different. | |
| May 19, 2018 at 17:44 | comment | added | Taras Banakh | @YCor By the way I also have learned something looking at Wiki on "adele" (=additive idele) and "idele" (= "ideal element"). Nice inventions! | |
| May 19, 2018 at 17:40 | comment | added | YCor | Yes, no problem! it made me learn (through Google search) of Chevalley's contribution to adeles and of Furstenberg's AMM note, so I'm happy with this :) | |
| May 19, 2018 at 17:39 | comment | added | Taras Banakh | @YCor I simply wanted to find some justifications for using the term "Furstenberg topology". In any case this was not my invention (I mean to call this topology Furstenberg), so maybe let us close this discussion. Thank you. | |
| May 19, 2018 at 17:18 | comment | added | YCor | @TarasBanakh yes I know, I have looked at the links before replying (and also did a little search about adeles). And I knew Furstenberg for various other things. Certainly while he was writing this prime number proof, he was aware of $p$-adic topologies and so on. (And I don't know why you're talking me about classical group theory. Chevalley's motivation with adeles was not group theory.) | |
| May 19, 2018 at 16:58 | comment | added | Taras Banakh | @YCor By the way, Furstenberg's area of specialization also was not classical Group Theory. The Furstenberg's topology arised as a trick for an unexpected topological proof of infinitude of prime numbers. The corresponding 1-page paper was published in Amer.Math.Monthly in 1955 and the proof is included to "The proofs from the Book". | |
| May 19, 2018 at 16:50 | comment | added | YCor | @TarasBanakh Indeed to which area (or rather community, since this is less subjectively defined) some mathematical object belongs is not obvious to decide. Norms (and thus topologies) on the integers play a fundamental role in arithmetic, initially not with a general-topology-focussed motivation. That such little pieces, which were not initially the main character, might have been rediscovered much later after first appearing, is quite frequent. | |
| May 19, 2018 at 16:45 | comment | added | YCor | (btw I messed up the @ in my previous comment, which was addressed to Taras, not Benjamin!) | |
| May 19, 2018 at 16:27 | comment | added | Taras Banakh | @YCor "This area" means a field of General Topology occupying with constructing various countable connected Hausdorff spaces. It started with the example of Urysohn in 20-ies and flourished in 70-ies (when a lot of various examples were constructed) and is still alive nowadays. Since it has relatively weak connection with Group Theory, the terminology is different. This probablly explains the term "Furstenberg topology". Another possible explanation is that Solomon Golomb that popularized Furstenberg and Golomb topologies wasn't professional mathematician but rather popularizer of mathematics. | |
| May 19, 2018 at 16:19 | comment | added | YCor | @BenjaminSteinberg it's a very standard topology on the integers, so I'm not sure what "this area" could mean. Adeles at least appear in a 1936 paper by Chevalley... | |
| May 19, 2018 at 16:14 | comment | added | YCor | @BenjaminSteinberg thanks, I indeed meant the pro-group one. | |
| May 19, 2018 at 15:35 | comment | added | Benjamin Steinberg | @YCor, this is the pro-group topology on the non-negative integers. The full profinite topology is discrete since it allows homomorphisms to any finite monoid not just fine groups. | |
| May 19, 2018 at 15:24 | vote | accept | Dominic van der Zypen | ||
| May 19, 2018 at 15:24 | comment | added | Taras Banakh | @YCor The term "Furstenberg topology" is standard and well-accepted in this area, see the seminal paper of Golomb (dml.cz/bitstream/handle/10338.dmlcz/700933/…) and the recent paper (alpha.math.uga.edu/~pete/CLLP_November_30_2017.pdf) | |
| May 19, 2018 at 15:22 | comment | added | Taras Banakh | @AlexM. Yes, you are right. Simply I do not remember all answers I give though I realize that this is very close to what was have been discussed earlier (especially the Golomb topology). I thought that the system shows all the related questions and answers at the moment of posing the question. So, the author always has a possibility to look and analyse what has already been asked and answered prior to posting his own question. | |
| May 19, 2018 at 15:18 | comment | added | Alex M. | Why do you answer the question instead of closing it as a duplicate of the other question where you yourself have given the accepted answer? | |
| May 19, 2018 at 15:04 | comment | added | YCor | Why do you call this "Furstenberg topology"? it seems it's plainly the profinite topology. | |
| May 19, 2018 at 14:24 | history | answered | Taras Banakh | CC BY-SA 4.0 |