Suslin operation

In mathematics, the Suslin operation 𝓐 is an operation that constructs a set from a collection of sets indexed by finite sequences of positive integers. The Suslin operation was introduced by and. In Russia it is sometimes called the A-operation after Alexandrov. It is usually denoted by the symbol 𝓐 (a calligraphic capital letter A).

Definitions
A Suslin scheme is a family $$P = \{ P_s: s \in \omega^{<\omega} \}$$ of subsets of a set $$X$$ indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set
 * $$\mathcal A P = \bigcup_{x \in {\omega ^ \omega}} \bigcap_{n \in \omega} P_{x \upharpoonright n}$$

Alternatively, suppose we have a Suslin scheme, in other words a function $$M$$ from finite sequences of positive integers $$n_1,\dots, n_k$$ to sets $$M_{n_1,..., n_k}$$. The result of the Suslin operation is the set
 * $$ \mathcal A(M) = \bigcup \left(M_{n_1} \cap M_{n_1, n_2} \cap M_{n_1, n_2, n_3} \cap \dots \right)$$

where the union is taken over all infinite sequences $$n_1,\dots, n_k, \dots$$

If $$M$$ is a family of subsets of a set $$X$$, then $$\mathcal A(M)$$ is the family of subsets of $$X$$ obtained by applying the Suslin operation $$\mathcal A$$ to all collections as above where all the sets $$M_{n_1,..., n_k}$$ are in $$M$$. The Suslin operation on collections of subsets of $$X$$ has the property that $$\mathcal A(\mathcal A(M)) = \mathcal A(M)$$. The family $$\mathcal A(M)$$ is closed under taking countable unions or intersections, but is not in general closed under taking complements.

If $$M$$ is the family of closed subsets of a topological space, then the elements of $$\mathcal A(M)$$ are called Suslin sets, or analytic sets if the space is a Polish space.

Example
For each finite sequence $$s \in \omega^n$$, let $$N_s = \{ x \in \omega^{\omega}: x \upharpoonright n = s\}$$ be the infinite sequences that extend $$s$$. This is a clopen subset of $\omega^\omega$. If $$X$$ is a Polish space and $$f: \omega^{\omega} \to X$$ is a continuous function, let $$P_s = \overline{f[N_s]}$$. Then $$P = \{ P_s: s \in \omega^{<\omega} \}$$ is a Suslin scheme consisting of closed subsets of $$X$$ and $$\mathcal AP = f[\omega^{\omega}]$$.