Souslin operation $\mathcal{\Gamma}$ contains countable union and intersection

Question

This is exercise 25.5 II) in Kechris Descriptive Set Theory:

Denoting by $\Gamma_\sigma$, $\Gamma_\delta$ the class of sets that are respectively countable unions or countable intersections of sets in $\Gamma$, show that if $X\in\Gamma$, then $\Gamma_\sigma\cup\Gamma_\delta\subseteq\mathcal{A}\Gamma$.

First off all I have a misunderstanding with the exercise, which might be some irrelevant concern on my end, so you make nothing wrong if you skip this part, if you are only interested in the actual question.

In which sense is $\Gamma$ a Souslin scheme/collection of subsets of a set $X$? The given definitions are as follows:

Let $(P_s)_{s\in\mathbb{N}^{<\mathbb{N}}}$ be a Souslin scheme on a set $X$, i.e., a familiy of subsets of $X$ indexed by $\mathbb{N}^{<\mathbb{N}}$. The Souslin operation $\mathcal{A}$ applied to such a scheme produces the set

$\mathcal{A}_sP_s=\bigcup_{x\in\mathcal{N}}\bigcap_{n} P_{x|n}$.

Given any collection $\Gamma$ of subsets of a set $X$ we denote by $\mathcal{A}\Gamma$ the class of sets $\mathcal{A}_sP_s$, where $P_s\subseteq X$ are in $\Gamma$.

So we just assume that $\Gamma$ is such a collection of subsets of a set $X$ (with the additional assumption that $X\in\Gamma$)? With other words $\Gamma\subseteq\mathrm{Pow}(X)$ (power set of $X$). Because the task does not specify this further. In particular which set it is a collection of subsets from. Anyways...

As far as the actual exercise goes:

I want to show $\Gamma_\sigma\cup\Gamma_\delta\subseteq\mathcal{A}\Gamma$.

To show $\Gamma_\delta\subseteq\mathcal{A}\Gamma$, I have to find for $G\in\Gamma_\delta$ a Souslin scheme $(P_s: s\in\mathbb{N}^{<\mathbb{N}})$, such that $\mathcal{A}P_s=G$.

$G$ is a countable intersection of elements in $\Gamma$, so it is most natural to take a Souslin scheme which contains the (countable many) sets which get intersected. I struggle alot to grasp what is happening here.

I need to associate an countable subset of $\Gamma$ (which consists of the sets which get intersected to get the set $G$) with an element $x$ of $\mathcal{N}=\mathbb{N}^{\mathbb{N}}$, this is my first issue. It seems I have to convert this into a Souslin scheme such that $\bigcap_n P_{x|n}=G$. This is the second issue.

Can you help me out? Thanks in advance.

Answer 1

If $B_0, B_1, B_2, \ldots \in \Gamma$ then $\bigcup_n B_n \in \mathcal{A}\Gamma$, and also $\bigcap_n B_n \in \mathcal{A}\Gamma$.

The $\Gamma$ is understood to be some collection of subsets of some $X$, so a subset of $\mathscr{P}(X)$, really. He does asume $X \in \Gamma$.

This can be seen by constructing "degenerate cases" of Suslin schemes:

For the intersection, set $A_s = B_{|s|}$, for all $s \in \Bbb N^{<\Bbb N}$, where $|s|$ denotes the length, as usual. For every $x \in \mathcal{N}$ the sequence $A_{s|n}$ just becomes the sequence $B_n$ and the whole union reduces to a single term $\bigcap_n B_n$.

For the union, we let the constant sequences (say with value $n$) in $\mathcal{N}$ give rise to an intersection equal to $B_n$ and all others only to at most subsets of that. So the union over $\mathcal{N}$ will be precisely $\bigcup_n B_n$.

E.g. set $A_s$ to be $B_{l(s)}$ for $s \in \Bbb N^{<\Bbb N}$, where $l(s)$ is the last term of $s$ (first term would also work BTW) and we also set $A_\emptyset = X$ (as a "neutral" first term in the intersection). As promised the constant $x \in \mathcal N$ just give rise to the set $B_n$, for that constant. So $\bigcup_n B_n \subseteq \mathcal{A}_s(A_s)$ and the reverse inclusion is immediate, so $\bigcup_n A_n \in \mathcal{A}\Gamma$.

So simple Souslin schemes give unions and intersections. In this case we don't "abuse" the option of taking actual uncountable unions.

Answer 2

First of all, $\Gamma$ is not a Suslin scheme. Your confusion arises from the use of the symbol $\mathcal{A}$ as a class operator: $\mathcal{A}\Gamma$ is the class of all sets obtained by applying the Suslin operation to a scheme $\{P_s\}$ with all $P_s$ in $\Gamma$.

Now to the question proper, intersections are immediate: Assume $Q_n$ for $n\in\mathbb{N_0}$ are sets in $\Gamma$. If you define that $P_s := Q_{|s|}$ (where $|s|$ is the length of $s$), then $\mathcal{A}_sP_s$ will equal $\bigcap_n Q_n$ (note that all the terms in the big union appearing in the $\mathcal{A}$ operation are the same).

By a similar reasoning, we can obtain the union of the $Q_n$, this time repeating a set in the intersections. Using the assumption that $X\in\Gamma$ allows for a slightly simpler solution. Define $$\begin{align*} P_{()} &:= X\\ P_s &:= Q_{s(0)}, \end{align*} $$ where $()$ denotes the empty sequence. This choice ensures that that all terms in the intersections inside the Suslin operation are equal and the final result is the countable union.