The Projective Hierarchy of finite rank

The Projective Hierarchy (finite rank)#

In full analogy to the definition of the boldface Borel hierarchy, but this time with \(\omega\) replaced by \(\mathcal{N}\). Define by recursion:

\[\begin{align*} \mathbf{\Sigma}^1_1 &= \exists^{\mathcal{N}}\mathbf{\Pi}^0_1,\\ \mathbf{\Pi}^1_n &= \neg\mathbf{\Sigma}^1_{n}, \\ \mathbf{\Sigma}^1_{n+1} &= \exists^\mathcal{N}  \mathbf{\Pi}^1_n\end{align*}\]

and put \(\mathbf{\Sigma}^1_n = \mathbf{\Sigma}^1_n \cap \mathbf{\Pi}^1_n\). The number \(n\) is called the rank of the pointclasses. The pointclasses \(\mathbf{\Sigma}^1_n\) are called the Lusin pointclasses, \(\mathbf{\Pi}^1_n\) are the dual Lusin pointclasses, and finally \(\mathbf{\Delta}^1_n\) are the ambiguous Lusin pointclasses.

The pointsets that occur in these Lusin pointclasses are called projective sets. In, again, full analogy to the case of the boldface Borel hierarchy, we have the usual hierarchy of pointclass inclusions for Lusin pointclasses. 

In the classical terminology the \(\mathbf{\Sigma}^1_1\) pointsets are called analytic or \(A\)-sets. They include most of the sets one encounters in hard analysis. The \(\mathbf{\Pi}^1_1\) pointsets are called coanalytic sets, or \(CA\)-sets. The \(\mathbf{\Sigma}^1_2\) pointsets are called \(PCA\)-sets, and we have \(CPCA\)-sets and so on.

Closure properties of Lusin pointclasses of finite rank#

Theorem. Each Lusin pointlass \(\mathbf{\Sigma}^1_n\ (n\geq 1)\) is closed under continuous substituion, \(\vee\),\(\wedge\),\(\forall^\leq\),\(\exists^\leq\),\(\forall^\omega\),\(\exists^\mathcal{N}\).

Theorem. Each dual Lusin pointclass \(\mathbf{\Pi}^1_n\ (n\geq 1)\) is closed under continuous substitution, \(\vee\), \(\wedge\), \(\forall^\leq\),\(\exists^\leq\),\(\exists^\omega\),\(\forall^\mathcal{N}\).

Theorem. Each ambiguous Lusin pointclass \(\mathbf{\Delta}^1_n\ (n\geq 1)\) is closed under continuous substituion, \(\neg\), \(\vee\),\(\wedge\),\(\exists^\leq\),\(\forall^\leq\),\(\exists^\omega\),\(\forall^\omega\).

The only difference between the proof of the the closure properties of Borel and Lusin pointclasses is that, in proving the closure properties of the latter, we need to code sequences of irrationals by single irrationals. Take a sequence of irrationals \(\{r_t\}_t\), then let \(\alpha\) be such a irrational that the \(k=\langle i,t\rangle\)-th component \((\alpha)_k = (r_t)_i\), where \(k=\langle i,t\rangle\) is the number that encodes the sequence \((i,t)\).

For easy comparison between the closure properties of Borel and Lusin pointclasses:

Theorem. Each Borel pointlass \(\mathbf{\Sigma}^0_n\ (n\geq 1)\) is closed under continuous substituion, \(\vee\),\(\wedge\),\(\forall^\leq\),\(\exists^\leq\),\(\exists^\omega\).

Theorem. Each dual Borel pointclass \(\mathbf{\Pi}^0_n\ (n\geq 1)\) is closed under continuous substitution, \(\vee\), \(\wedge\), \(\forall^\leq\),\(\exists^\leq\),\(\forall^\omega\).

Theorem. Each ambiguous Borel pointclass \(\mathbf{\Delta}^0_n\ (n\geq 1)\) is closed under continuous substituion, \(\neg\), \(\vee\),\(\wedge\),\(\exists^\leq\),\(\forall^\leq\).