Timeline for Unique limits of sequences plus what implies Hausdorff?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 16, 2021 at 23:23 | comment | added | Alec Rhea | You may already be aware of this, but unique convergence of nets is equivalent to Hausdorff in any topological space. | |
| Sep 13, 2011 at 4:19 | answer | added | Paul Fabel | timeline score: 12 | |
| Sep 12, 2011 at 14:15 | answer | added | Dave L Renfro | timeline score: 4 | |
| Sep 7, 2011 at 15:31 | comment | added | David White | Looking back at my notes, it wasn't Frechet spaces, but rather the Frechet topology on any space $X$, where you define $A$ to be closed iff $A$ is the set of limits of sequences in $A$. This topology can be used to construct an example of an anti-Hausdorff space with unique limits, but the answer below is just as good. | |
| Sep 7, 2011 at 15:28 | answer | added | David White | timeline score: 7 | |
| Sep 7, 2011 at 15:23 | comment | added | David White | It seems you're right about Frechet spaces. I admit that I was just quoting a line from a topology course I took some time ago, and I didn't stop to question whether or not it's true. However, the cocountable one is right and I'll post the details as an answer | |
| Sep 7, 2011 at 13:46 | comment | added | Dirk | I'm a bit confused. The Wikipedia page says that Frechet spaces are indeed Hausdorff. Also: Why do you have unique limits of sequences in the cocountable topology? | |
| Sep 7, 2011 at 13:41 | history | edited | Dirk | CC BY-SA 3.0 |
Corrected link
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| Sep 7, 2011 at 13:39 | vote | accept | Dirk | ||
| Sep 7, 2011 at 12:50 | comment | added | David White | I believe having unique limits implies the space is $T_1$, so perhaps the question boils down to $T_1$ plus what implies $T_2$. | |
| Sep 7, 2011 at 12:45 | comment | added | David White | The link in your question doesn't work for me. Anyway, it's easy to produce non-Hausdorff spaces with unique limits. The Frechet Topology has unique limits and is anti-Hausdorff (all open sets intersect). Another example is the co-countable topology on $\mathbb{R}$. See en.wikipedia.org/wiki/Fr%C3%A9chet_space and en.wikipedia.org/wiki/Cocountable_topology | |
| Sep 7, 2011 at 9:41 | answer | added | Michael Greinecker | timeline score: 20 | |
| Sep 7, 2011 at 9:21 | history | asked | Dirk | CC BY-SA 3.0 |