Timeline for Is the lexicographic ordering on the unit square perfectly normal?
Current License: CC BY-SA 4.0
12 events
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| Jan 30, 2020 at 13:58 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Jan 30, 2020 at 13:57 | comment | added | Will Brian | @RobertFurber: You're perfectly right, of course. A perfectly normal space is a normal space in which every closed set is $G_\delta$. (I was thinking of this whole problem in the context of normal spaces, but apparently that's not clear. I'll edit to make it clear.) | |
| Jan 30, 2020 at 3:03 | comment | added | Robert Furber | In the the Moore plane/Niemytskii plane every closed set is $G_\delta$, but it is not normal. Call me old-fashioned, but I think "perfectly normal" should imply "normal". It is true, via a proof using Urysohn's lemma countably many times and carefully summing up the resulting functions, that a perfectly normal space is the same thing as a normal space in which every closed set is $G_\delta$. Of course, this does not affect the argument for the lexicographic ordering not being normal, because only the direction "perfectly normal $\Rightarrow$ all closed sets are $G_\delta$" is used. | |
| Jan 29, 2020 at 12:54 | comment | added | Will Brian | @NateEldredge: Good observation! It's not exactly using a nuke to kill a fly, but it is using more than just the flyswatter. I've edited to simplify things. | |
| Jan 29, 2020 at 12:53 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Jan 28, 2020 at 16:58 | vote | accept | VDGG | ||
| Jan 28, 2020 at 16:58 | vote | accept | VDGG | ||
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| Jan 28, 2020 at 15:54 | comment | added | Nate Eldredge | Just to note a slight variation (since I had been thinking about a similar proof before I saw yours): the complement of $A_U$ is actually at most countable, since every point of it is isolated in the usual topology of $[0,1]$. So any $G_\delta$ containing $C$ must in fact contain co-countably many vertical lines, and you don't need Baire. | |
| Jan 28, 2020 at 15:51 | comment | added | Nate Eldredge | I think you missed one: "horizontal line" at the end should be "vertical line"? | |
| Jan 28, 2020 at 15:48 | history | edited | Will Brian | CC BY-SA 4.0 |
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| Jan 28, 2020 at 15:44 | vote | accept | VDGG | ||
| Jan 28, 2020 at 16:58 | |||||
| Jan 28, 2020 at 15:27 | history | answered | Will Brian | CC BY-SA 4.0 |