Timeline for Partitioning $\mathbb{R}$ into $\aleph_1$ Borel sets
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Feb 21, 2012 at 20:28 | comment | added | Goldstern | A slightly more explicit version of Andreas' argument: First, identify reals with trees in some Borel way. Here, trees are subsets of $\omega^{< \omega}$ which are downward closed. Next, each tree $T$ has a derivative $T'$: remove all leaves. This naturally defines a transfinite sequence of iterated derivatives $(T^{(\alpha)}: \alpha< \omega_1)$. Let the ("Cantor-Bendixson") rank of $T$ be the first $\alpha<\omega_1$ such that $T^{(\alpha)}$ has no leaves (possibly, but not necessarily, because it is empty). Now the set of trees with a given rank is certainly Borel. | |
| Aug 6, 2011 at 20:31 | comment | added | François G. Dorais | Thanks Andreas! This is much more constructive than I thought it would be. | |
| Aug 6, 2011 at 20:26 | vote | accept | François G. Dorais | ||
| Aug 6, 2011 at 20:09 | history | answered | Andreas Blass | CC BY-SA 3.0 |