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Is there a bijection solution $\eta:\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R}$ (with an explicit formula, if it is possible) for the inequality $$\min\{x,y\}\leq \eta(x,y)\leq \max\{x,y\}\;\; ; \;\; x\neq y?$$

Martin Sleziak
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M.H.Hooshmand
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  • A bijection with explicit formula.... I'd say it's quite hopeless. – Vim May 07 '17 at 07:45
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    It probably should not be too difficult to show that such function exists, some construction base on transfinite induction seems like a natural approach. Writing down a specific function with these properties seems to be more difficult. – Martin Sleziak May 07 '17 at 08:48
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    Perhaps it is worth mentioning that there are some posts on this site about an explicit bijection between $\mathbb R^2$ and $\mathbb R$. For example, this question and perhaps also some of the posts linked there. However, the additional condition on minimum and maximum makes this more complicated. – Martin Sleziak May 07 '17 at 08:56
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    You have asked this question twice: why don't you delete one of the copies (the previous one, for instance)? – Alex M. May 07 '17 at 09:47
  • The old one is in the sense of set theoretic aspect – M.H.Hooshmand May 07 '17 at 10:22

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