Projective hierarchy

Remarks

The coanalytic sets form $\Pi^1_1$, and by Suslin's theorem $\Delta^1_1$ is the class of Borel sets. Example: the set of $x \in \set{0, 1}^{\N \times \N}$ that code a well-order of $\N$ is in $\Pi^1_1$ but not in $\Sigma^1_1$. Non-example by counting: there are only $2^{\aleph_0}$ projective subsets of $\R$, and $2^{2^{\aleph_0}}$ subsets in all, so some subset of $\R$ is not projective. Each class $\Sigma^1_n$ and $\Pi^1_n$ is closed under countable unions, countable intersections and preimages by continuous maps, and $\Sigma^1_n \cup \Pi^1_n \subseteq \Delta^1_{n+1}$. If $X$ is uncountable, then $\Sigma^1_n \neq \Pi^1_n$ for every $n$. The lightface version, defined with recursive codes instead of arbitrary parameters, is part of the analytical hierarchy.