Is there a bijective map from $[0,1]$ to the unit sphere, (or to all vectors in $\mathbb{R}^{3}$ with length $1$ about $(0,0,0)$), ideally with an even distribution such that $[0,0.5)$ or $]$, maps to half the volume of the sphere etc.
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4They have the same cardinality, so yes. – Jose Avilez Jan 10 '22 at 00:56
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Related: https://math.stackexchange.com/questions/183361/examples-of-bijective-map-from-mathbbr3-rightarrow-mathbbr – Bonnaduck Jan 10 '22 at 00:57
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There is even a Borel bijection. If you normalize measures so that both the interval and the sphere have the same total measure, then there is a measure-preserving Borel bijection. – Andreas Blass Jan 10 '22 at 04:08