Is it true that for every uncountable set $U$, we can write $U=V∪W$, where $V$ and $W$ are disjoint uncountable subsets of $U$ ?
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As I remarked in the comment on MO. This was asked before. – Asaf Karagila May 07 '13 at 16:26
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1Also see my answer here. – Asaf Karagila May 07 '13 at 16:30
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(In fact I urge the next person who votes to close as a duplicate to use the link from my second comment). – Asaf Karagila May 07 '13 at 16:32
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@AsafKaragila: Thanks a lot. – Cold Play May 07 '13 at 16:32
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1Please search the site the next time you are told that this was asked before. – Asaf Karagila May 07 '13 at 16:32
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Yes, assuming the axiom of choice. Without the axiom of choice one might have an amorphous set. Such a set is uncountable in the sense that it is not finite, and there is no bijection between it and $\Bbb N$. However, it is not the disjoint union of any two infinite subsets.
Brian M. Scott
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