Timeline for Direct construction of the Stone-Čech compactification using ultrafilters?
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
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| Dec 18, 2010 at 21:38 | comment | added | Daniel Litt | This certainly only works for normal spaces. | |
| Dec 18, 2010 at 14:14 | comment | added | Greg Graviton | Found a slightly different equivalence relation that only mentions the topology of $X$. The trick is too choose $\sim$ large enough to identify convergent ultrafilters while making it small enough to separate ultrafilters that can be distinguished by continuous functions into compact spaces. | |
| Dec 18, 2010 at 14:10 | history | edited | Greg Graviton | CC BY-SA 2.5 |
New equivalence relation
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| Dec 18, 2010 at 9:09 | comment | added | Greg Graviton | What do you mean? The quantification over all compact spaces $C$ is fairly natural and the proof goes through. But you seem to be unhappy with it, probably because it's "too indirect", i.e. it would be nicer if only the space $X$ were mentioned? It's clear that any two convergent ultrafilters need to be equal in the quotient; the problem is to identify ultrafilter which are not convergent in $X$ but do converge in all compact images of $X$. | |
| Dec 18, 2010 at 1:50 | comment | added | Qiaochu Yuan | Yes, this is essentially what I was thinking. To prove continuity and the universal property it should be easier to prove that all the relevant maps preserve convergence of ultrafilters. But the essential problem, it seems to me, is the definition of \sim, and I will edit my question accordingly. | |
| Dec 17, 2010 at 11:13 | history | edited | Greg Graviton | CC BY-SA 2.5 |
Language
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| Dec 17, 2010 at 11:09 | comment | added | Greg Graviton | Fair enough. I have completely rewritten my answer. | |
| Dec 17, 2010 at 11:08 | history | edited | Greg Graviton | CC BY-SA 2.5 |
Proper answer
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| Dec 16, 2010 at 18:52 | comment | added | Qiaochu Yuan | Right, but this presupposes the existence of the Stone-Čech compactification in general. I'm trying to use this argument to construct the Stone-Čech compactification in general. | |
| Dec 16, 2010 at 16:59 | history | edited | Greg Graviton | CC BY-SA 2.5 |
Fix type of inclusion
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| Dec 16, 2010 at 11:26 | history | answered | Greg Graviton | CC BY-SA 2.5 |