Transfinite Borel Pointclasses

Preliminary: Countable operations#

Definition (countable pointset operation). A function \(\Phi\) with domain some set of infinite sequences of pointsets and pointsets as values is called a countable pointset operation. We use the notation that is similar to Einstein's summation convention, that is, \(\Phi_i P_i := \Phi(P_0,P_1,P_2,\ldots)\). For a pointclass \(\Lambda\), \(\Phi \Lambda\) is defined as the set \(\{\Phi_i P_i : P_i \in \Lambda\text{ and }\Phi_i P_i\text{ defined}\}\).

Definition (closure). A pointclass \(\Lambda\) is closed under a countable operation \(\Phi\) whenever if \(P_i\in \Lambda\) for all \(i\) and \(\Phi_i P_i\) is defined, \(\Phi_i P_i \in \Lambda\).

A family of examples of countable operations is countable conjunctions/disjunctions, \(\wedge^\omega, \vee^\omega\), with their obvious meaning. Set theoretically, whenever all the \(P_i\) are subsets of the same space, we have \(\wedge_i^\omega P_i = \cap_i P_i\) and \(\vee^\omega_i P_i = \cup_i P_i\).

For pointclasses, there are relations between countable quantifiers and countable conjunction/disjunctions. The following theorem is a version of Hebrand's theorem

Theorem. Let \(\Gamma\) be and \(\mathcal{N}\)-parametrized pointclass which is closed under continuous substituion. If \(\Gamma\) is closed under \(\exists^\omega\), then it is closed under \(\vee^\omega\). If \(\Gamma\) is closed under \(\forall^\omega\), then it is closed under \(\wedge^\omega\).

Corollary. Each \(\mathbf{\Sigma}^0_n\) is closed under \(\vee^\omega\) and each \(\mathbf{\Pi}^0_n\) is closed under \(\wedge^\omega\). All \(\mathbf{\Sigma}^1_n, \mathbf{\Pi}^1_n, \mathbf{\Delta}^1_n\) are closed under both \(\vee^\omega\) and \(\wedge^\omega\).

For a pointclass \(\Lambda\), if \(\Lambda\) is closed under continuous substitution, it is easy to see that \(\exists^\omega\Lambda \subseteq \vee^\omega\Lambda\), which essentially means that every projection along \(\omega\) of a set in \(\Lambda\) can be written as a countable union of sets in \(\Lambda\).

Extending the Borel pointclasses to transfinite ranks#

We have seen that each \(\mathbf{\Sigma}^0_n\) is closed under \(\vee^\omega\) and each \(\mathbf{\Pi}^0_n\) is closed under \(\wedge^\omega\), and that \(\exists^\omega\Lambda\ subseteq \vee^\omega \Lambda\). This allows for a new inductive characterization of the finite-rank Borel pointclasses:

\[\begin{align*}\mathbf{\Sigma}^0_1 &= \text{all open sets.}\\\mathbf{\Sigma}^0_{n+1} &= \vee^\omega \neg \mathbf{\Sigma}^0_n.\end{align*}\]

The class of all Borel pointsets of finite rank is closed under \(\exists^\omega\) but not under \(\vee^\omega\): e.g. let \(G_n\subseteq \mathcal{N}\) to be in \(\mathbf{\Sigma}^0_n \setminus \mathbf{\Pi}^0_n\) and verify that \(G := \cup_n\{(n,\alpha):\alpha\in G_n\}\) is not in any \(\mathbf{\Sigma}^0_n\). We extend the finite Borel hierarchy into the transfinite as follows. For each ordinal number \(\xi > 1\): put

\[\mathbf{\Sigma}^0_\xi = \vee^\omega \neg (\cup_{\eta <\xi}\mathbf{\Sigma}^0_\eta).\]

This means that \(P \in \mathbf{\Sigma}^0_\xi\) if there are pointsets \(P_0, P_1,\ldots\) with each \(P_i\) in some \(\mathbf{\Sigma}^0_\eta, \eta<\xi\), s.t. \(P = \cup_i(\mathcal{X}\setminus P_i)\). The dual and the ambiguous Borel pointclasses are then defined in the obvious ways.

Definition (Borel sets). The pointclass \(\mathsf{B} = \cup_xi \mathbf{\Sigma}^0_\xi\) is called the pointclass of Borel sets. The members of the pointclass \(\mathsf{B}\) are called Borel sets.

We provide a characterization of Borel sets. 

Theorem. For each product space \(\mathcal{X}\), the class \(\mathsf{B}\restriction \mathcal{X}\) of Borel subsets of \(\mathcal{X}\) is the smallest collection of subsets of \(\mathcal{X}\) which contains the open sets and is closed under complementation and countable union. Similarly, \(\mathsf{B}\restriction\mathcal{X}\) is the smallest collection of subsets of \(\mathcal{X}\) which contains the open (or the closed sets) and is closed under countable union and countable intersection.