Timeline for Theorems for nothing (and the proofs for free)
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 14, 2023 at 23:07 | comment | added | Noah Schweber | Several years late, but here's a proof from the Monthly: jstor.org/stable/2319896 | |
| Feb 21, 2010 at 12:04 | comment | added | Philip Brooker | I also like the fact that every countable compact Hausdorff space is homeomorphic to a countable successor ordinal equipped with its order topology. This is the Mazurkiewicz-Sierpinski theorem, published originally in French (I think) but also available in English in Z. Semadeni's book 'Banach spaces of continuous functions' in section 8 (the chapter on compact 0-dimensional spaces). A proof of the Alexandroff-Hausdorff theorem (i.e., every compact metric space is a continuous image of the Cantor set) is also there, as well as a bunch of other tasty topology. | |
| Dec 2, 2009 at 14:36 | comment | added | Mark Meckes | After this theorem has done its job changing your intuition, though, it's pretty easy to believe. A surjective continuous map glues stuff together. And the Cantor set is not "just" a sprinkling of dust; it's a sprinkling of a whole lot of rather clumpy dust. So it shouldn't be surprising that you can glue all this clumpy dust into many different forms. | |
| Nov 14, 2009 at 0:25 | comment | added | Tom Leinster | Right! Once you know it, it's fine. But I think it's capable of changing one's intuition on what spaces and maps are. After all, the Cantor set is just a sprinkling of dust; how could it be capable of covering a big fat space like the 3-ball? | |
| Nov 13, 2009 at 19:04 | comment | added | Harald Hanche-Olsen | Surprising, yes, but once you know about it, it seems easy enough to cook up a proof. Just write the set as a union of two closed subsets, decide to map the left half of the Cantor set onto one and the right half to the other, then do the same to each of these two sets, and so on. In the limit you have the map you want, provided you have arranged for the diameters of the parts to go to zero. | |
| Nov 13, 2009 at 17:58 | comment | added | Tom Leinster | There's a book by Willard and another by Hocking and Young, both called something like "Topology" or "General Topology". But I can't remember whether they specifically cover THIS theorem, because there are two closely related other theorems about the Cantor set, C: (1) every totally disconnected compact metric space is homeomorphic to a subset of C; (2) every totally disconnected compact MS with no isolated points is either empty or homeo to C. Those books cover at least one of these three theorems. | |
| Nov 13, 2009 at 16:59 | comment | added | Justin DeVries | That's quite surprising! What are some good references for this? | |
| Nov 13, 2009 at 16:42 | history | answered | Tom Leinster | CC BY-SA 2.5 |