Timeline for Is there a general theory of "compactification"?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Oct 15, 2019 at 10:32 | comment | added | Henno Brandsma | @RobertFurber That only holds because that ordinal space is “almost compact” to begin with, in quite a strong sense : it has a unique uniformity inducing its topology. | |
| Oct 15, 2019 at 1:06 | comment | added | Robert Furber | @R.vanDobbendeBruyn That is a common line of thinking, based on familiar examples such as $\mathbb{N}$ and $\mathbb{R}$. But consider the fact that if you give $\aleph_1$ its order topology, its Stone-Čech compactification is the same as its 1-point compactification. | |
| Oct 14, 2019 at 23:05 | comment | added | Terry Tao | In part this is because the universal objects tend not to exist within algebraic geometry. For instance the universal cover of a complex variety tends to be transcendental in nature and outside the algebraic geometry "universe"; instead one has to take formal inverse limits of finite extensions to work with things like the etale fundamental group as a substitute. Similarly any Stone-Cech type "universal blowup" of an algebraic singularity would be extremely non-algebraic in nature, so one works with one or more finitely complicated partial blowups instead. | |
| Oct 14, 2019 at 16:03 | comment | added | R. van Dobben de Bruyn | To me it seems that Stone–Čech has to be big because it tries to be universal. On the other hand, one-point compactifications and compactifications in algebraic geometry try to be nice spaces. | |
| Oct 14, 2019 at 15:33 | history | answered | Andreas Blass | CC BY-SA 4.0 |