Timeline for Is there a continuous partition of space into circles?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 29 at 21:06 | comment | added | Alexey Ustinov | Related question $\mathbb{R}^3$ as the union of disjoint circles. In particular it contains a link to the earlier article. | |
| Oct 25 at 14:07 | comment | added | Alessandro Codenotti | @JoelDavidHamkins I had not seen your question, but unfortunately it went unanswered... I think this kind of questions are extremely interesting | |
| Oct 25 at 13:08 | comment | added | Joel David Hamkins | @AlessandroCodenotti Indeed, I asked exactly that question here on MathOverflow 12 years ago: mathoverflow.net/q/93601/1946. Can we prove there is no Borel partition of space into unit circles? And similarly for the other kinds of partitions. It still has no answer. | |
| Oct 25 at 10:29 | comment | added | Alessandro Codenotti | I have often wondered and chatted with people about the least complexity of a partition of $\Bbb R^3$ into circles (as a subspace of the Vietoris hyperspaces $K(\Bbb R^3)$). For example is there a Borel partition of the space into circles of unit radius? This is probably related to whether you can produce such a thing without AC. What is the least complexity a partition into circles of different sizes can have? One can ask many similar related questions | |
| Oct 24 at 22:37 | answer | added | Moishe Kohan | timeline score: 17 | |
| Oct 24 at 22:14 | history | became hot network question | |||
| Oct 24 at 15:10 | comment | added | Sam Nead | @RyanBudney - when the parts of the partition are just continuous loops, how do you prove that the quotient space is an orbifold? | |
| Oct 24 at 15:08 | comment | added | Gabe K | One quick observation is that if you consider the one point compactification of $\mathbb{R}^3$ (i.e. $\mathbb{S}^3$), then Hopf fibration does the job. | |
| Oct 24 at 14:52 | answer | added | Sam Nead | timeline score: 19 | |
| Oct 24 at 14:48 | comment | added | Joel David Hamkins | @RyanBudney Sorry, I don't really know what that means. Could you post an answer explaining the argument? | |
| Oct 24 at 14:47 | comment | added | Ryan Budney | Your continuity requirement would force $\mathbb R^3$ to be a circle bundle, i.e. the total space of a bundle with fiber a circle, and it is not. | |
| Oct 24 at 14:35 | comment | added | Francois Ziegler | I guess Borel & Serre’s Impossibilité de fibrer un espace euclidien par des fibres compactes should be relevant… | |
| Oct 24 at 14:10 | comment | added | Joel David Hamkins | Please help me retag if appropriate. | |
| Oct 24 at 14:09 | history | asked | Joel David Hamkins | CC BY-SA 4.0 |