Timeline for Continuous images of $\beta \mathbb{N} \setminus\mathbb{N}$
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| May 15, 2018 at 2:06 | vote | accept | Claudia Correa | ||
| Mar 21, 2018 at 21:55 | comment | added | Not Mike | @YCor effectively yes, that's what I'm saying, see Theorem 3.2 of projecteuclid.org/euclid.ijm/1255454110 | |
| Mar 21, 2018 at 21:49 | comment | added | YCor | But only theoretically... e.g., if I have "given" spaces $X,Y$, I don't see if the problem whether $X'$ and $Y'$ are homeomorphic is trivial... | |
| Mar 21, 2018 at 21:47 | comment | added | YCor | @NotMike I'm not sure what you're proving (I'm not familiar with all notation). Are you saying that to every compact space $X$ one can canonically associate some totally disconnected space $X'$ with some quotient map $X'\to X$ (this Gleason extremally disconnected space of $X$?) such that $X$ is a quotient of $bN-N$ iff $X'$ is a quotient of $bN-N$? Indeed I'd then agree that this more precise assertion makes the problems at least theoretically equivalent. | |
| Mar 21, 2018 at 21:33 | comment | added | Not Mike | @YCor whoops meant $CO(G)$ instead of $G$. | |
| Mar 21, 2018 at 21:08 | comment | added | Not Mike | @YCor let $B$ be the completion of $A=\mathcal{P}(\omega)/fin$ then any continuous map taking $S(A)$ onto $X$ can be pulled-back to a map of $S(B)$ onto $X$. Let $G$ be the extermally disconnected Gleason space associated with $X$; since extremally disconnected compact spaces are projective, either $G$ maps onto $S(B)$ or vice versa; without loss of generality, $S(B)$ maps onto $G$, in this case we can find some $A_0 \subset A$ which generates a sub-algebra isomorphic to a dense sub-algebra of $G$. | |
| Mar 21, 2018 at 20:48 | comment | added | YCor | @NotMike I don't think that the fact you mention make the problems equivalent. | |
| Mar 21, 2018 at 20:11 | comment | added | Not Mike | @YCor Since every compact space is the continuous image of an extremally disconnected compact space, it is enough to only consider totally disconnected images; the two problems are equivalent. | |
| Mar 21, 2018 at 17:11 | comment | added | YCor | In restriction to totally disconnected spaces, this is equivalent, by Stone duality, to classifying Boolean subalgebras of $2^\mathbf{N}/(\mathrm{fin})$ up to isomorphism. Here $(\mathrm{fin})$ is the ideal of finite subsets. | |
| Mar 21, 2018 at 13:38 | history | edited | Will Brian | CC BY-SA 3.0 |
deleted 1 character in body; edited title
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| Mar 21, 2018 at 2:19 | history | edited | Johannes Hahn | CC BY-SA 3.0 |
TeXified title and question
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| Mar 21, 2018 at 1:49 | history | edited | Martin Sleziak |
added (stone-cech) tag
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| Mar 21, 2018 at 1:40 | answer | added | Will Brian | timeline score: 23 | |
| Mar 21, 2018 at 0:49 | review | First posts | |||
| Mar 21, 2018 at 2:19 | |||||
| Mar 21, 2018 at 0:46 | history | asked | Claudia Correa | CC BY-SA 3.0 |