Timeline for Can $\mathbb{R}$ be written as an uncountable union of disjoint uncountable subsets?
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| May 30, 2023 at 9:50 | comment | added | Sassatelli Giulio | About your side question on the cardinality of $B$ (of course with AC), for a cardinal $\kappa$ the following are equivalent: 1) there is a subfield $F\subseteq \Bbb R$ such that $\dim_F(\Bbb R)=\kappa$; 2) $\kappa=1\lor \aleph_0\le \kappa\le \lvert \Bbb R\rvert$. | |
| May 30, 2023 at 9:28 | comment | added | Dave L. Renfro | For some decompositions that go a bit beyond purely set-theoretic constructions (in size and/or in "naturalness"), see my answer to Using transfinite induction to split $R$ to continuum many pairwise disjoint subsets of $R$ AND A partition of the unit interval into uncountably many dense uncountable subsets AND my answer to Are there sets of zero measure and full Hausdorff dimension?. | |
| May 30, 2023 at 8:33 | comment | added | Anne Bauval | Hint: $\mathbb{R}^2$ has the cardinality of the continuum. | |
| May 30, 2023 at 8:26 | comment | added | Anne Bauval | Your $B$ may be a singleton. | |
| May 30, 2023 at 8:15 | history | asked | Ifielmodes | CC BY-SA 4.0 |