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I was thinking about this:

$[0,1]$ can be partitioned into a countable union of uncountable sets. Write $[0,1]=(0,1]\cup \{0\}:$

$$(0,1]=\bigcup_{n=1}^{\infty}\Big(\frac{1}{n+1},\frac{1}{n}\Big]$$

$[0,1]$ can be partitioned into an uncountable union of countable sets. Write $[0,1]=(0,1]\cup \{0\}:$

$$(0,1]=\bigcup_{x\,\in \,(1,2]}\Big\{\frac{x}{2^n}:\;n\in\mathbb{N}\Big\}$$

I cannot find a partitioning of $[0,1]$ and into an uncountable union of uncountable sets though. Does one exist? I was maybe thinking of taking a union of Cantor-like sets, but maybe I've missed an easy one.

user118224
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  • These partition questions always get tons of votes and answers before anyone bothers to check if they are duplicates... – Asaf Karagila Jan 03 '14 at 18:31

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