Timeline for Action of the permutation group on the set of topologies on a continuum
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 27, 2021 at 21:57 | comment | added | LSpice | @JoelDavidHamkins, thanks for the correction. | |
| Apr 27, 2021 at 19:26 | vote | accept | Kolao | ||
| Apr 27, 2021 at 19:14 | answer | added | YCor | timeline score: 3 | |
| Apr 27, 2021 at 19:04 | comment | added | Joel David Hamkins | @LSpice Your claim that any two subsets of cardinality continuum will be in bijection is not quite right, since you need the complements also to be equinumerous, in order to have a bijection of the whole space. But it doesn't affect your point, since there will be plenty of instances where this is true. | |
| Apr 27, 2021 at 19:03 | comment | added | Joel David Hamkins | Ah, that is clearly not true, as @LSpice mentions. (I tried to save you!) | |
| Apr 27, 2021 at 19:00 | comment | added | Kolao | @JoelDavidHamkins No I meant that any individual topology defining a manifold is fixed by the action. | |
| Apr 27, 2021 at 18:57 | comment | added | YCor | Every finite orbit is reduced to a singleton ($X$ arbitrary infinite set), since then $S(X)$ has no nontrivial finite quotient. The question is for which topologies on $X$, the group of self-homeomorphism consists of all permutations. I think it's not too hard to answer. | |
| Apr 27, 2021 at 18:57 | comment | added | Joel David Hamkins | But meanwhile, the set of all manifolds is fixed set-wise by the action, since any permutation of the underlying set is a (homeomorphic) manifold. Perhaps that is what the OP meant? The set of instances of any truly topological property will be fixed in this sense. | |
| Apr 27, 2021 at 18:53 | comment | added | LSpice | For a fixed point, in fact, open-ness depends only on the cardinalitty, so there is a minimum cardinality of a non-empty open set, and the open sets are precisely the empty set and those that have at least that cardinality. | |
| Apr 27, 2021 at 18:52 | comment | added | LSpice | The topology of a manifold is very much not a fixed point; for example, on $\mathbb R$, any two subsets of the cardinality of the continuum will be in bijection, whether they are open intervals, closed intervals, or anything else. | |
| Apr 27, 2021 at 18:42 | review | First posts | |||
| Apr 27, 2021 at 18:51 | |||||
| Apr 27, 2021 at 18:41 | history | asked | Kolao | CC BY-SA 4.0 |