Intereting Posts

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The book Theory of Functions of a Real Variable by I. P. Natanson, proves that a denumerable or finite union of pairwise disjoint sets of cardinality $c$ has cardinality $c$.

The proofs given in the book are fairly easy, using the axiom of choice. But then it is left as an exercise a generalization of the above theorems for not necessarily pairwise disjoint sets.

I have searched all over the web, but all the proofs found so far are for pairwise disjoint sets.

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- Does any uncountable set contain two disjoint uncountable sets?

The proof given in the book relies on the fact that each set $A_k$ can be matched with a set like $[a_{k-1},a_k)$, with $a_k$ point of a partition of $[0,1]$, such that $\bigcup_{k=1}^{\infty\lor n}A_k=[0,1)$. So it is easy to guess the bijective function, but I can’t generalize this proof.

Any help is highly appreciated. Thanks and regards.

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HINT: Let $B_1=A_1$. For $n\ge 2$ let $B_n=A_n\setminus\bigcup_{k=1}^{n-1}A_k$. Then $\bigcup_nB_n=\bigcup_nA_n$, the sets $B_n$ are pairwise disjoint, each has cardinality at most $\mathfrak{c}$, and $B_1$ has cardinality $\mathfrak{c}$.

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