I am working on measurable sets and I have been coming across this, so called "fact" about unit circle. More precisely, some of proofs I am studying based on the following observation:
There is a Borel set in the unit square whose projection onto the first coordinate is not a Borel set in the interval $[0,1]$.
I can't give a proof or visual image of this set. I thought two- dimensional Cantor sets i.e., $\mathcal{C}\times \mathcal{C}$.
On
The projection of a Borel set in the square onto the $x$-axis is a so-called analytic set or Souslin set in the line. So you are asking for an analytic set in the interval that is not a Borel set. That is, an analytic set whose complement is not analytic. Any text on descriptive set theory should have such a construction. For example:
Donald L. Cohn, Measure Theory (Birkhäuser, 1980) Corollary 8.2.17, page 269
There is an analytic subset of $\mathcal N$ that is not a Borel set.
Here, $\mathcal N$ is homeomorphic to the irrationals in $(0,1)$, for example.
As a consequence, any uncountable complete metric space has such a set as well.
You are not alone in finding this tough, as Lebesgue himself believed (in 1905) he had proved that the projection of a Borel set in $\mathbb{R}^2$ onto one of the axes was a Borel set. His error was discovered by Souslin which lead (I believe) to the process of Souslin schemes/hierarchy that he developed to investigate this problem.
R. M. Dudley "Real Analysis and Probability"
In my oldish edition it is covered in chapter 13, although my edition is by Chapman and Hall, whereas the Amazon edition is published by Cambridge. I am not sure that he actually gives a proof of the existence of such a set, but it is a good place to start, and he gives lots of references at the end of each chapter.
There is also another (older) book by Rogers on "Analytic Sets", but I don't have a copy so I can't check it.