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11:06 AM
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A: Projection of Borel set from $R^2$ to $R^1$

Andrey RekaloThis was a famous mistake made by Lebesgue (see also Gerald Edgar's answer to this MO question). Suslin showed that a plane Borel set exists whose projection is not a Borel set. See the references to the original article by Suslin and related works here.

The link to eom.springer.de is broken, but the article can now be found at encyclopediaofmath.org/wiki/Suslin_theorem. — The Amplitwist Jul 24 at 12:01
@TheAmplitwist The question was bumped by another edit - so I have edited this answer too. (It's up to you whether you want to keep the comment or remove it now.)
 
11:35 AM
@Martin Thanks for letting me know, I opted to remove my comment.
 

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