Timeline for Compactness of symmetric power of a compact space
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2021 at 18:08 | answer | added | Steven Clontz | timeline score: 2 | |
| Aug 23, 2021 at 8:35 | comment | added | Alessandro Codenotti | To add to Z.M.'s comment you don't need to assume $G$-invariance of the metric, if $d$ is not $G$-invariant it is another exercise to show that $d'(x,y)=\sum_g d(gx,gy)$ is an equivalent $G$-invariant metric. (This generalizes to actions of compact groups by integrating against the Haar measure along orbits) | |
| Aug 22, 2021 at 20:18 | comment | added | Sunrit | Thank you so much! | |
| Aug 22, 2021 at 16:03 | review | Close votes | |||
| Aug 27, 2021 at 3:04 | |||||
| Aug 22, 2021 at 12:46 | comment | added | Tom Goodwillie | I don't have a reference, but I think that what Z.M. asserts is a straightforward exercise. | |
| Aug 22, 2021 at 8:08 | comment | added | Alessandro Codenotti | Another way to think about this is that the quotient is the orbit space with the Hausdorff metric | |
| Aug 22, 2021 at 5:27 | comment | added | Z. M | Following Tom Goodwillie's comment, I think that the following is true: let $X$ be a metric space with a $G$-action where $G$ is a finite group and the metric is $G$-invariant. Then the orbit space $X_G$ is metrizable with $d([x],[y])=\min\{d(x,gy)\vert g\in G\}$ (check that "open" balls are really open and constitute a neighborhood basis). Maybe we could generalize this to compact groups $G$. | |
| Aug 22, 2021 at 1:35 | comment | added | Sunrit | Hi @TomGoodwillie thank you for the response. Can you give some references/insight into why? | |
| Aug 22, 2021 at 1:18 | comment | added | Tom Goodwillie | The topology on $\mathcal X/\sim$ given by the metric you describe is the same as the quotient topology. | |
| Aug 21, 2021 at 22:57 | review | First posts | |||
| Aug 22, 2021 at 5:53 | |||||
| Aug 21, 2021 at 22:47 | history | asked | Sunrit | CC BY-SA 4.0 |