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Q: Can every uncountable subset $\mathbb{R}$ be split at some number into two parts of the same cardinality?

Vladimir Reshetnikov Is it possible to prove without Continuum Hypothesis that for every uncountable subset $S$ of $\mathbb{R}$ there is a real number $x$ that splits it into two parts of the same cardinality, i.e. $\left|S \cap (-\infty,x)\right|=\left|S \cap (x,\infty)\right|$? (if the answer to the first questio...

 

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