Timeline for Importance of separability vs. second-countability
Current License: CC BY-SA 3.0
22 events
| when toggle format | what | by | license | comment | |
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| Jan 11, 2022 at 5:32 | answer | added | Cloudscape | timeline score: 0 | |
| Dec 24, 2019 at 12:09 | answer | added | Santi Spadaro | timeline score: 2 | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Aug 2, 2015 at 13:36 | comment | added | Todd Trimble | @PaulTaylor Please let us know (either here or at the nForum: nforum.ncatlab.org) if you have spotted an error in the nLab article. | |
| Jun 5, 2013 at 20:21 | vote | accept | The User | ||
| May 21, 2013 at 21:42 | answer | added | Ramiro de la Vega | timeline score: 12 | |
| May 19, 2013 at 14:59 | comment | added | Paul Taylor | Thank you, Andrej, but you took the photo! I am a long way from being able to discuss separability and second countability, but for a published introduction to overtness see A Lambda Calculus for Real Analysis www.paultaylor.eu/ASD/lamcra or if you are keen to see an incomplete attempt to introduce the idea to "ordinary mathematicians" you could look at Overt Subspaces of $R^n$ www.paultaylor.eu/drafts/overtrn.pdf . I am not sure that the nLab account is correct. | |
| May 19, 2013 at 12:42 | comment | added | Andrej Bauer | And this would be Paul Taylor: paultaylor.eu The photo is a bit blurry, but that's how I percieve Paul most of the time anyway. | |
| May 19, 2013 at 12:41 | comment | added | Andrej Bauer | Overtness is exactly as complicated as compactness. nLab has a bit written about it: ncatlab.org/nlab/show/overt+space | |
| May 19, 2013 at 8:24 | comment | added | user5810 | As far as I could tell, he means something that is incredibly complicated. $\:$ | |
| May 19, 2013 at 7:44 | comment | added | The User | What do you mean by “overtness”? Openness? And who is Paul Taylor? | |
| May 19, 2013 at 7:07 | comment | added | Andrej Bauer | This might not be the right decade to make this comment, but separability seems to be just the poor man's version of overtness, something Paul Taylor has been pointing out. If this is indeed the case, then separability would indeed be a suboptimal notion. | |
| May 19, 2013 at 6:16 | comment | added | user5810 | Ahem. $\:$ If countable choice then second countability is a stronger condition than separability. $\;\;$ | |
| May 19, 2013 at 5:59 | answer | added | Joseph Van Name | timeline score: 11 | |
| May 19, 2013 at 0:56 | comment | added | Włodzimierz Holsztyński | Isn't thee theorem about separability of the product of continuum many separable spaces due to Edward Szpilrajn-Marczewski? BTW, it holds for the product of $2^a$ spaces with dense subsets of cardinality less or equal $a$. | |
| May 18, 2013 at 19:22 | comment | added | François G. Dorais | Garrett Birkhoff's undergraduate thesis was on "Axiomatic Definitions of Perfectly Separable Metric Spaces" ams.org/journals/bull/1933-39-08/S0002-9904-1933-05693-8/… | |
| May 18, 2013 at 19:16 | comment | added | François G. Dorais | Apparently, perfectly separable and completely separable are synonyms of 2nd-countable. | |
| May 18, 2013 at 19:06 | comment | added | François G. Dorais | This 1935 reference also uses this convention: ncbi.nlm.nih.gov/pmc/articles/PMC1076540/pdf/pnas01754-0040.pdf | |
| May 18, 2013 at 19:03 | comment | added | François G. Dorais | Hm. I deleted my comment before I saw yours. | |
| May 18, 2013 at 18:22 | comment | added | The User | @François That is wrong—I have implicitly given an example above: The product of continuum many copies of the circle group is a (locally) compact group which is separable (like every product of at most continuum many separable spaces, according to a theorem by Ross and Stone, jstor.org/stable/2313241), but not even first-countable (it is counterexample 105 in Counterexamples in Topology). | |
| May 18, 2013 at 18:13 | answer | added | Nik Weaver | timeline score: 7 | |
| May 18, 2013 at 17:05 | history | asked | The User | CC BY-SA 3.0 |