Timeline for Direct construction of the Stone-Čech compactification using ultrafilters?
Current License: CC BY-SA 2.5
12 events
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| Dec 18, 2010 at 3:21 | history | edited | Daniel Litt | CC BY-SA 2.5 |
added 286 characters in body
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| Dec 18, 2010 at 3:20 | comment | added | Daniel Litt | Ah you're right, I assumed the existence of a locally countable base. I'll edit to fix this. | |
| Dec 18, 2010 at 3:11 | comment | added | Qiaochu Yuan | @Daniel: I don't think you can conclude that every element of the closure of f(X) in C is the limit of a convergent sequence without countability hypotheses. What is true is that every element of the closure is the limit of an ultrafilter, so we can replace X^N with the cardinality of the set of ultrafilters on X (much larger, but still a bound). | |
| Dec 18, 2010 at 3:01 | history | edited | Daniel Litt | CC BY-SA 2.5 |
Addressed Qiaochu's edit 2.
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| Dec 18, 2010 at 1:54 | comment | added | Daniel Litt | And I think you're right about Stone-Cech factoring through the "complete regularification," though I don't have a proof off the top of my head. | |
| Dec 18, 2010 at 1:53 | comment | added | Daniel Litt | Well again, KHaus certainly doesn't see "arbitrary subsets" so there's some heuristic reason to think that no natural functor will be built from the power set. In any case there is a natural generalization of ultrafilters to arbitrary algebras of sets; the version of an ultrafilter you have in mind is the specialization to the case where the algebra of sets is just the power set. Walker's book has a pretty good exposition as I recall, but alas I don't have it on hand. | |
| Dec 18, 2010 at 1:47 | comment | added | Qiaochu Yuan | Yep. Okay, so if I understand what you're saying, it's basically that the version of Stone-Čech I want factors through the "completely regularification" of X. That makes sense. But the notion of ultrafilter I'm working with involves arbitrary subsets of X, not closed sets, and it seems to work fine; I still don't see the need for the latter notion. | |
| Dec 18, 2010 at 1:27 | history | edited | Daniel Litt | CC BY-SA 2.5 |
Explained why KHaus loves "zero sets"; edited body
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| Dec 18, 2010 at 0:59 | comment | added | Daniel Litt | Ah sorry, didn't see the comment. So what this boils down to, I think, is that you don't require this version of Stone-Cech to be fully faithful? | |
| Dec 18, 2010 at 0:23 | comment | added | Qiaochu Yuan | As I explained in the comments I am only looking for a left adjoint to the inclusion of the category of compact Hausdorff spaces into the category of topological spaces; I'm not requiring that the induced map be an embedding. I've heard of the zero-set construction but I don't see why it should be necessary. | |
| Dec 17, 2010 at 19:38 | history | edited | Daniel Litt | CC BY-SA 2.5 |
added 348 characters in body
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| Dec 17, 2010 at 19:32 | history | answered | Daniel Litt | CC BY-SA 2.5 |