Timeline for Can continuous correspondence be represented via continuous functions?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Feb 25, 2023 at 8:14 | comment | added | Emily | I've also written a bit about this here | |
| Feb 25, 2023 at 8:12 | comment | added | Emily | @Ded As Michael says, given a correspondence (=relation) $R\colon X⇸Y$ you can put the upper or lower Vietoris topology in $\mathcal{P}(Y)$ and then define continuitiy/openness/closedness of $R$ via its adjunct $R^\dagger\colon X\to\mathcal{P}(Y)$. A reference for openness/closedness in this context is Clementino–Tholen, p. 9, while one for upper/lower semicontinuity for total relations is Klein–Thompson's Theory of Correspondences, Theorems 7.1.4 and 7.1.7. | |
| Feb 24, 2023 at 19:02 | vote | accept | Ded | ||
| Feb 24, 2023 at 18:55 | answer | added | Michael Greinecker | timeline score: 2 | |
| Feb 24, 2023 at 18:37 | comment | added | Michael Greinecker | Upper and lower hemicontinuity make sense without a topology on sets. That being said, the Vietoris topology does the job. | |
| Feb 24, 2023 at 18:26 | comment | added | Ded | I do not have enough intuition to think of correspondences as mappings from sets to collections of sets so I am not sure what an appropriate topology would be. | |
| Feb 24, 2023 at 18:23 | comment | added | Iosif Pinelis | The discrete topology (which is of course not Euclidean in any sense) would be quite bad, as almost no map from a set with the Euclidean topology to a set with the discrete topology can be continuous. | |
| Feb 24, 2023 at 18:16 | comment | added | Ded | That should be the discrete topology which is given as the collection of all subsets of $\mathcal{X}$. | |
| Feb 24, 2023 at 18:02 | comment | added | Iosif Pinelis | What is that "Euclidean" topology on the set of all subsets of $\mathcal X$? Can you describe it formally? | |
| Feb 24, 2023 at 18:00 | comment | added | Ded | Euclidean as well! | |
| Feb 24, 2023 at 17:58 | comment | added | Iosif Pinelis | What topology do you have on the set of all subsets of $\mathcal X$? | |
| Feb 24, 2023 at 17:50 | comment | added | Ded | w.r.t. euclidean topology. | |
| Feb 24, 2023 at 17:46 | comment | added | Iosif Pinelis | "$C$ is continuous" ... With respect to what topology? | |
| Feb 24, 2023 at 17:43 | comment | added | Ded | Oops, should be fixed now! | |
| Feb 24, 2023 at 17:43 | history | edited | Ded | CC BY-SA 4.0 |
added 18 characters in body
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| Feb 24, 2023 at 17:42 | comment | added | Iosif Pinelis | What is $\theta$? If $\theta\in\Theta$, then what does $f_1(\theta,x)$ mean, given that $f$ is defined on $\Theta$? | |
| Feb 24, 2023 at 17:31 | comment | added | Ded | made a notation change, hope it is clearer now! | |
| Feb 24, 2023 at 17:30 | history | edited | Ded | CC BY-SA 4.0 |
Fixed notation
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| Feb 24, 2023 at 17:25 | comment | added | Ben McKay | What does $f(\theta,x)\ge 0$ mean for $f(\theta,x)\in\mathbb{R}^d$? | |
| Feb 24, 2023 at 17:17 | history | asked | Ded | CC BY-SA 4.0 |