I have a problem concerning the output of (and the intuition behind) the Suslin $\mathcal{A}$-Operation.
More specifically, I really don't see exactly what the output of it really is (even if I can actually use it to prove some very basic things), and I think this hampers my general understanding.
In order to get some intuition, I decided to translate everything in terms of finite sets (this one should be an instantiation of the Alexandroff Operation). By doing this, I also added some numbered questions to emphasise my problems.
This is the basic setting:
$X = \{ a, b \}$
$A = \{0, 1 \}$
$A_e = \{ a, b \}$, $A_0 = \{ a \}$, $A_1 = \{ b \}$, $A_{00} = \varnothing = A_{10} = A_{01} = A_{11}$
where a Suslin scheme is a function of the form $A : A^{<\mathbb{N}} \to 2^X$.
So, we have that
$$ \mathcal{A}_2 ( \{ A_s \} ) = \bigcup_{\alpha \in A^{\mathbb{N}}} \bigcap_{n \in \mathbb{N}} A_{\alpha|n} \hspace{7cm} (\square),$$
where $\{ A_s \}$ denotes the family of all the $A$ sets previously considered (this is a Suslin scheme that is also regular), and $\alpha|n$ denotes the initial segment $\alpha = ( \alpha_0, \alpha_1 , \dots, \alpha_{n-1})$ of length $n$ of a $\alpha \in A^{\mathbb{N}}$. Hence we have that
$$x \in \mathcal{A}_2 ( \{ A_s \} ) \Longleftrightarrow \exists \alpha \in A^{\mathbb{N}} : \forall n \in \mathbb{N} \ ( x \in A_{\alpha|n}) \hspace{3cm}(*).$$
Now, my problems start here.
1. Does this actually means that in this case, with finite $X$, the $\mathcal{A}$-Operation gives an empty result?
This is my hypothesis because my intuition is the following. We are in a finite context, and no matter what $n \in \mathbb{N}$ we pick, still the process basically ends at the second digit (i.e. $A_{**}$) which is associated with the empty set, and hence there is no element that can fit the RHS of $(*)$.
2. Is this line of reasoning correct?
If this is the case, then a natural consideration is that the power of the Suslin $\mathcal{A}$-Operation really comes from infinite spaces, where this process can go on indefinitely.
3. Again, is this correct?
I am really looking forward to any feedback.
Thank you in advance for your time.
PS: The notation I used should be rather standard in set theory, but I will clarify if needed.
In order to have a Suslin scheme, you must have a set $A_\sigma$ for each $\sigma\in{^{<\omega}X}$; since you have only $A_\sigma$ for $|\sigma|\le 2$, you don’t have a Suslin scheme and cannot define the result of the $\mathscr{A}$-operation. If you set $A_\sigma=\varnothing$ for all $\sigma\in{^{<\omega}X}$ of length greater than $1$, so as to have a genuine Suslin scheme, then for each $\varphi\in{^\omega\omega}$ you have
$$\bigcap_{n\in\omega}A_{\varphi\upharpoonright n}=\varnothing\;,$$
since already $A_{\varphi\upharpoonright 2}=\varnothing$. Thus,
$$\bigcup_{\varphi\in{^\omega\omega}}\bigcap_{n\in\omega}A_{\varphi\upharpoonright n}=\varnothing$$
as well.