Timeline for maximum with respect inclusion of a function whose output are sets
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2010 at 21:45 | comment | added | Pietro Majer | @alberto: " $S$ is a topological space": do you have a precise topology in mind? (just " any topology " would not make a difference, as Damiano pointed out) | |
| Aug 25, 2010 at 19:40 | comment | added | damiano | @alberto: this is not going to be enough to make the question interesting, as the only open sets of $S$ could be $S$ itself and the empty set. I suspect that, if the question were properly formulated, then the answer would be "no", and compactness would play a key role in finding a "fixed point". Unfortunately, I am not able to guess what the right question should be. | |
| Aug 25, 2010 at 18:20 | comment | added | alberto | I could add another hypothesis. I assume that the space of sets S is a topological space and A->S is continuous | |
| Aug 25, 2010 at 18:05 | comment | added | alberto | Yes, this example is fine. Thank you | |
| Aug 25, 2010 at 17:52 | comment | added | damiano | So compactness of $A$ was not relevant? In any case, is the example above fine for you? I was trying to find a "natural" example for which $W$ was continuous and iterations of $W$ had no finite orbits, since a finite orbit gives you immediately the consequence you were asking to avoid. Obviously there are easier examples: for instance, just do the same as in my answer, but only considering a single orbit, so that $A=\mathbb{Z}$, the function $W$ is the shift by 1 and $S(n) = \left\{m \in \mathbb{Z} \,|\, m \leq n \right\}$. Topologize $\mathbb{Z}$ as you want, if you need to. | |
| Aug 25, 2010 at 17:26 | comment | added | alberto | In reality no, there is no required property for the space of sets S. | |
| Aug 25, 2010 at 16:31 | comment | added | damiano | Btw, I realize now that compactness was not used in the argument above: this is suspicious... Presumably you also wanted some property of the "space of sets $S$" and of the function $A \to S$ so that compactness would play a role, right? | |
| Aug 25, 2010 at 16:28 | history | answered | damiano | CC BY-SA 2.5 |