Timeline for infinite permutations
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 10, 2010 at 8:53 | comment | added | Pete L. Clark | @JDH: no, you are not missing anything. As I said, I was missing the explicit connection between the OP's "limit" and a topology on the set of permutations, which did not appear in your answer. As you say, a few minutes' thought shows that the permutations with finite support are dense in $\operatorname{Sym}(X)$. Because I was meeting a friend for dinner, I didn't want to be a few minutes late, so the comment I posted was not very insightful. It happens... | |
| Mar 10, 2010 at 8:38 | vote | accept | kakaz | ||
| Mar 10, 2010 at 1:24 | comment | added | Joel David Hamkins | Pete, I don't quite understand your worry. Am I missing something? The basic open sets in this topology are determined by the action of a permutation on a finite set, and clearly any such action that is compatible with a permutation is compatible with a finite permutation, so the finite permuations are dense. Also, the finite approximations of the product of transpositions in my answer clearly converge to the whole permutation, so one can see that every permutation is a limit of finite permutations this way. From my perspective, this question has been answered now several times over. | |
| Mar 10, 2010 at 0:28 | comment | added | Pete L. Clark | Is the subset of permutations with finite support dense in $\operatorname{Sym}(X)$ in this topology? (I have to go to dinner, or I would think about this myself.) Note that Gjergi comes at the problem in a different way, essentially forcing this to be the case by taking the profinite completion. | |
| Mar 10, 2010 at 0:26 | comment | added | Pete L. Clark | I feel like the answers to question 2. are not yet complete, because of the following point: can the notion of limit of finite permutations be construed with respect to a natural topology on $\operatorname{Sym}(X)$? One natural topology on this set is the subset of the compact-open function space topology on all functions from $X$ to $X$, with $X$ given the discrete topology. I believe that this is also the topology of "pointwise convergence", i.e., such that a sequence of permutations $\sigma_n$ converges iff for all $x \in X$, $\sigma_n(x)$ is eventually constant. | |
| Mar 9, 2010 at 23:58 | answer | added | Douglas Zare | timeline score: 7 | |
| Mar 9, 2010 at 23:14 | answer | added | Gjergji Zaimi | timeline score: 6 | |
| Mar 9, 2010 at 21:38 | answer | added | Tony Huynh | timeline score: 2 | |
| Mar 9, 2010 at 21:32 | answer | added | Joel David Hamkins | timeline score: 10 | |
| Mar 9, 2010 at 20:55 | vote | accept | kakaz | ||
| Mar 9, 2010 at 20:59 | |||||
| Mar 9, 2010 at 20:54 | answer | added | Michael Lugo | timeline score: 2 | |
| Mar 9, 2010 at 20:29 | history | asked | kakaz | CC BY-SA 2.5 |