Timeline for Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 11, 2022 at 22:14 | comment | added | Saúl RM | Yeah this proof was written a bit too fast as I was in the middle of an exam week. I have tried to write it in more detail in MSE | |
| Jun 11, 2022 at 18:59 | comment | added | Tereza Tizkova | Thank you! I am still quite unsure, after consulting that with colleagues, so I tried to rephrase your answer to the initial question on stackexchange and gave you credits. I am not sure if you have account there, but I highlighted what is still unclear and will appreaciate if you take a look. math.stackexchange.com/questions/4468793/… | |
| Jun 10, 2022 at 21:42 | comment | added | Saúl RM | The procedure always gives $U$ as a union of clopens, the problem is that only finitely many of them may be nonempty (for example this happens if $A_1=U$). However this cannot happen if $U$ is not contained in any finite union of the $A_n$, because any finite union of the clopens we obtain is contained in a finite union of the sets $A_n$, so it can't be the whole $U$ | |
| Jun 10, 2022 at 16:45 | comment | added | Tereza Tizkova | Thank you, this is wonderful! The only part I dont understand is how the "This implies that $U\equiv\mathbb{Q}^\infty$..." follows from that we get infinitely many clopens if we pick $A_n$ so that no finite union covers $U$? How do we "get infinite many non empty clopens"? Thank you | |
| Jun 10, 2022 at 9:30 | vote | accept | Tereza Tizkova | ||
| Jun 9, 2022 at 17:36 | history | edited | Saúl RM | CC BY-SA 4.0 |
added 465 characters in body
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| Jun 9, 2022 at 17:30 | history | answered | Saúl RM | CC BY-SA 4.0 |