Counting the number of partitions of $\mathbb{R}$ into countable subsets

Question

I'm trying to determine the number of partitions of $\mathbb{R}$ into countable subsets. Obviously each partition will contain uncountably many sets and the cardinality of the set of all such partitions is bounded above by the number of relations on $\mathbb{R}$ (which is $2^c$) but I haven't gotten any further than that. Thanks for your help!

Answer 1

The number of such partitions is $2^{\mathfrak c}$.

First, each partition $\pi$ is a collection of disjoint subsets of $\mathbb R$, so it has size at most $\mathfrak c$, and so there are at most $|[\mathcal P(\mathbb R)]^{\le\mathfrak c}|=2^{\mathfrak c}$ partitions, where $[A]^{\le\kappa}$ denotes the collection of subsets of $A$ of size at most $\kappa$.

Second, split $\mathbb R$ as the disjoint union $A\cup B\cup C$, each set of size $\mathfrak c$. Fix a partition $\{D_x:x\in C\}$ of $A$, and a partition $\{E_x:x\in C\}$ of $B$, where each $D_x$ and each $E_x$ is countably infinite. Given $F\subseteq C$, consider the partition $\rho_F$ of $\mathbb R$ into countably infinite sets given by the sets $D_x$ for $x\notin F$, $D_x\cup\{x\}$ for $x\in F$, $E_x\cup\{x\}$ for $x\notin F$, and $E_x$ for $x\in F$. The assignment $F\mapsto \rho_F$ is injective, and therefore there are at least $2^{\mathfrak c}$ partitions of $\mathbb R$ into countably infinite sets.