I'm studying Jech's Set Theory and I noticed that he doesn't show an example of a non analytic set. Of course the analytic sets are Lebesgue measurable, have the Baire property and the perfect set property, so I know a few obvious examples.
Can you show me a less ugly subset of $\mathbb R$ that is not the projection of a closed set in $\mathbb R^2$? (I know the definition was with a Borel set in $X\times\mathcal N$ but $\mathbb R^2$ is easier to work with, so...)
My guess is that $\mathbb R\setminus\mathbb Q$ may not be analytic but I have no proof for it.
If I remember correctly you can find good examples of non-analytic sets in Kechris book $\textit{Classical Descriptive Set Theory}$. For example the set $WO$ of codes of countable ordinals is $\boldsymbol\Pi^1_1$-complete. Also, say a linear order $(A,<)$ is scaterred if there is no order preserving map from of $(\mathbb{Q},<)$ into $(A,<)$. Then let $x \in SCAT \leftrightarrow x$ codes a linear order and $A_x$ is scattered. Then $SCAT$ is $\boldsymbol\Pi^1_1$-complete. A last nice example (still in Kechris and due to Marzurkiewickz) is that the set of all differentiable function in $C([0,1])$ is $\boldsymbol\Pi^1_1$-complete (and so is the set of continuous functions with everywhere convergent Fourier series, Atjai-Kechris). I am sure there are plenty of examples in the higher pointclasses as there is a whole industry devoted to this nice subject of finding sets of objects and computing their exact complexity.