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 = \$ of subsets of a set $X$ indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set
:$\mathcal A P = \bigcup_ \bigcap_ P_$

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 $M_$. The result of the Suslin operation is the set
:$\mathcal A(M) = \bigcup \left(M_ \cap M_ \cap M_ \cap \dots \right)$
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 $M_$ 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 intersections and—if $X\in M$—countable unions, 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 = \$ be the infinite sequences that extend $s$.
This is a clopen subset of $\omega^\omega$.
If $X$ is a Polish space and $f: \omega^ \to X$ is a continuous function, let $P_s = \overline$.
Then $P = \$ is a Suslin scheme consisting of closed subsets of $X$ and \mathcal AP = f omega^/math>.

References

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*{{citation, first=M. Ya., last= Suslin, journal= C. R. Acad. Sci. Paris , volume= 164 , year=1917, pages= 88–91, title=Sur un définition des ensembles measurables ''B'' sans nombres transfinis

Descriptive set theory