Timeline for Are $\beta \mathbb{Q}$ and $\beta(\beta\mathbb{Q}\setminus\mathbb{Q})$ homeomorphic?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 7, 2023 at 8:09 | vote | accept | Jakobian | ||
| May 5, 2023 at 20:40 | history | edited | LSpice | CC BY-SA 4.0 |
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| May 5, 2023 at 15:40 | answer | added | Jakobian | timeline score: 7 | |
| May 5, 2023 at 15:08 | comment | added | Jakobian | @Anonymous Ah I see! If $x$ is in the remainder of the Stone-Cech compactification of $\beta\mathbb{Q}\setminus\mathbb{Q}$, then the character at that point is uncountable by above argument. But if $x\in \beta\mathbb{Q}\setminus \mathbb{Q}$, then since character of a point is the same for any dense embedding, it must be equal to $\chi(x, \beta\mathbb{Q})$ which again by above argument is uncountable. | |
| May 5, 2023 at 13:42 | comment | added | Anonymous | I don't know where this is explicitly stated--it might be in the Walker book on the Stone-Cech compactification--but it is not hard. If $p$ is a point of first countability of $\beta X \setminus X$, then $\beta X \setminus \{p\}$ is $\sigma$-compact and therefore normal. But if $a = (a_n)$ a sequence in $X$ that converges to $p$, then $\{a_n: n = 1, 2, ...\}$ is closed in $X$ but a function which alternates between $0$ and $1$ on that set does not extend to $p$. | |
| May 5, 2023 at 13:03 | comment | added | Jakobian | @Anonymous do you have a reference? | |
| May 5, 2023 at 11:51 | comment | added | Anonymous | $\beta \mathbb{Q}$ has a dense metrizable subset (and points of first countability) whereas $\beta (\beta {\mathbb{Q}} \setminus {\mathbb{Q}})$ has neither. | |
| May 5, 2023 at 0:18 | comment | added | Joseph Van Name | For each partition of $\beta\mathbb{Q}$ into two components $A,B$, one of those components has a non-trivial convergent sequence. I don't see the same thing happening with $\beta(\beta\mathbb{Q}\setminus\mathbb{Q})$ but I am currently too busy to write and check the proof. | |
| May 4, 2023 at 21:56 | history | edited | Jakobian | CC BY-SA 4.0 |
rewrote what I wrote previously for better clarity
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| May 4, 2023 at 21:42 | history | edited | Jakobian | CC BY-SA 4.0 |
added 1045 characters in body
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| May 4, 2023 at 21:35 | comment | added | Jakobian | @Gro-Tsen Yes, see edit2 | |
| May 4, 2023 at 21:34 | history | edited | Jakobian | CC BY-SA 4.0 |
added 1045 characters in body
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| May 4, 2023 at 21:18 | comment | added | Gro-Tsen | Can you justify or provide a reference for the statement of the first sentence? Is it supposed to be a well-known or trivial fact? | |
| May 4, 2023 at 19:58 | history | edited | Jakobian | CC BY-SA 4.0 |
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| May 4, 2023 at 19:20 | comment | added | R. van Dobben de Bruyn | At a first glance, it seems to me they might have wildly different cardinalities. Or did you try this already? | |
| May 4, 2023 at 15:41 | history | asked | Jakobian | CC BY-SA 4.0 |