Timeline for Subsets of the Cantor set
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 7, 2022 at 18:00 | history | edited | LSpice | CC BY-SA 4.0 |
Deleted "thanks"
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| Mar 7, 2022 at 17:41 | history | edited | gaam2296 | CC BY-SA 4.0 |
added 9 characters in body
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| Mar 7, 2022 at 17:40 | vote | accept | gaam2296 | ||
| Mar 7, 2022 at 17:40 | answer | added | gaam2296 | timeline score: 1 | |
| Mar 7, 2022 at 11:46 | review | Close votes | |||
| Mar 13, 2022 at 3:11 | |||||
| Mar 7, 2022 at 11:25 | answer | added | KP Hart | timeline score: 3 | |
| Mar 5, 2022 at 21:17 | comment | added | YCor | @SaúlRodríguezMartín oops, my mistake, yes of course: if closed (=clopen) nonempty it's homeomorphic to Cantor, and if not closed it's homeomorphic to Cantor minus singleton. Anyway this leaves the reasoning unchanged: it works in any Hausdorff $U'$ with no isolated point. | |
| Mar 5, 2022 at 21:11 | comment | added | Saúl RM | @YCor just wanted to point out that $U$ can also be homeomorphic to the Cantor set. Also, the same reasoning can be applied without knowing how open subsets of the Cantor set are up to homeomorphism: it is enough to notice that $U$ is uncountable if it is non empty (because every nonempty open set of the Cantor set contains a small copy of the Cantor set), and every point of $U$ is an accumulation point of $D\cap U$. | |
| Mar 5, 2022 at 21:01 | comment | added | YCor | If $U$ is assumed non-empty, just being a nonempty open subset of a Cantor set, it is homeomorphic to Cantor minus one point (or equivalently to Cantor $\times$ discrete countable). So the point is to check that every dense subset $D$ of $U$ has infinitely many accumulation points inside $D$. Indeed every element of $D$ is an accumulation point of $D$ (this just follows from the fact that $U$ has no isolated point). | |
| Mar 5, 2022 at 20:59 | comment | added | YCor | If $U$ is empty the assertion is false. So it should be assumed that $U$ is nonempty. | |
| S Mar 5, 2022 at 20:45 | review | First questions | |||
| Mar 5, 2022 at 21:05 | |||||
| S Mar 5, 2022 at 20:45 | history | asked | gaam2296 | CC BY-SA 4.0 |