Timeline for When (why) did we allow manifolds to be non-Hausdorff and/or non-second countable?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 6, 2020 at 18:29 | comment | added | Praphulla Koushik | @DmitriPavlov I understand.. This is the second reference to same "red herring" :D | |
| Jun 6, 2020 at 18:24 | comment | added | Dmitri Pavlov | @PraphullaKoushik: Yes. He was merely responding to an inquiry about the meaning of the term “topological space with a smooth atlas” that confused it with something else. You cannot make inferences about acceptable terminology from such a clarifying remark. See also ncatlab.org/nlab/show/red+herring+principle about this. | |
| Jun 6, 2020 at 17:48 | comment | added | Praphulla Koushik | @DmitriPavlov In the comment below his answer he says "I just meant to say, any smooth manifold, without the requirement that it is 2nd countable, Hausdorff, paracompact... just a bare atlas."... Did I misunderstand something? | |
| Jun 6, 2020 at 17:41 | comment | added | Dmitri Pavlov | @PraphullaKoushik: I think you obviously misread David Carchedi's answer, which says “Whether one takes manifolds to mean 2nd countable + Hausdorff, or whether one removes these conditions and considers all topological spaces with a smooth atlas…”. As you can see, he uses the term “topological space with a smooth atlas” and does not mention the term “manifold” at all when he talks about non-Hausdorff or nonparacompact objects. | |
| Jun 5, 2020 at 13:58 | comment | added | user21349 | I suppose this depends on who is included in "we." It doesn't include relativists. For a discussion, see Earman, pitt.edu/~jearman/Earman2008a.pdf . The question should probably be "who (why)" rather than "when (why)." | |
| Jun 5, 2020 at 7:14 | comment | added | Adrian Clough | One elementary reason for contemplating non-second countable manifolds is that it allows you to view any set, not just countable ones, as a discrete manifold. If one replaces second countable with paracompact, any connected component of a manifold is still second countable. | |
| Jun 4, 2020 at 17:58 | answer | added | Timothy Chow | timeline score: 8 | |
| Jun 4, 2020 at 14:01 | history | became hot network question | |||
| Jun 4, 2020 at 12:23 | comment | added | Praphulla Koushik | @DmitriPavlov It is easy to blame anything on Algebraic geometry :P It has all kind of surprising properties/set up/constructions.. :D | |
| Jun 4, 2020 at 9:17 | answer | added | Stefan Waldmann | timeline score: 8 | |
| Jun 4, 2020 at 9:01 | answer | added | guest | timeline score: 10 | |
| Jun 4, 2020 at 8:35 | comment | added | Praphulla Koushik | @DmitriPavlov oh, I did not knew (do not know yet) about C^\infty rings.. | |
| Jun 4, 2020 at 7:37 | comment | added | Dmitri Pavlov | The Zariski topology on the maximal spectrum of a C^∞-ring is Hausdorff, so this example is not relevant. | |
| Jun 4, 2020 at 7:03 | answer | added | Dmitri Pavlov | timeline score: 11 | |
| Jun 4, 2020 at 5:57 | comment | added | Praphulla Koushik | Is it because of the influence of algebraic geometry (where the very first topological space (Zariski topology) we come across is non-Hausdorff)? | |
| Jun 4, 2020 at 5:56 | history | asked | Praphulla Koushik | CC BY-SA 4.0 |