Timeline for Uniqueness of limits and compactness implies closure
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 22, 2018 at 0:01 | answer | added | Henno Brandsma | timeline score: 3 | |
| Dec 13, 2018 at 18:37 | comment | added | Nate Eldredge | @user44191: Yeah, presumably. I'm just going off Wikipedia here; haven't tried to check it myself. | |
| Dec 13, 2018 at 18:31 | comment | added | user44191 | @NateEldredge Do you mean something like the "open" Tychonoff plank (i.e. with both edges removed) plus two corners? Otherwise, I'm missing how the corner isn't the limit of an edge sequence. | |
| Dec 13, 2018 at 18:19 | comment | added | Nate Eldredge | @user44191: That suggests looking at a Tychonoff plank with two corners; apparently the corner is a limit point, but not the limit of any ordinal-indexed sequence. On the other hand, if you say "ultrafilter limits are unique" or "net limits" then this is equivalent to being Hausdorff. | |
| Dec 13, 2018 at 18:14 | comment | added | user44191 | I'd suggest the following as a possible replacement: uniqueness of limits of any cardinal sequence, rather than only countable ones. | |
| Dec 13, 2018 at 18:03 | comment | added | Nate Eldredge | I think it's necessary, though, because a convergent sequence together with one of its limits is a compact set, and it is closed only if it contains all the limits of the sequence. | |
| Dec 13, 2018 at 18:01 | comment | added | Nate Eldredge | It's not going to be sufficient, because sequence convergence isn't enough to detect "limits". You should be able to get a counterexample from a "line with two origins" construction on the uncountable ordinal $\omega_1 + 1$. | |
| Dec 13, 2018 at 17:11 | comment | added | Daniel Elessar | Yes, when I say uniqueness of limits I mean the property of a topological space that every convergent sequence converges uniquely. | |
| Dec 13, 2018 at 17:10 | comment | added | user44191 | When you say "uniqueness of limits", do you mean the statement that if a sequence converges, then it converges uniquely? Alternatively, please explain the precise statement you are referring to. | |
| Dec 13, 2018 at 17:00 | review | First posts | |||
| Dec 13, 2018 at 17:10 | |||||
| Dec 13, 2018 at 16:59 | history | asked | Daniel Elessar | CC BY-SA 4.0 |