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 formula_1 of subsets of a set formula_2 indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set 
Alternatively, suppose we have Suslin scheme, in other words a function formula_4 from finite sequences of positive integers formula_5 to sets formula_6. The result of the Suslin operation is the set 
where the union is taken over all infinite sequences formula_8
If formula_4 is a family of subsets of a set formula_2, then formula_11 is the family of subsets of formula_2 obtained by applying the Suslin operation formula_13 to all collections as above where all the sets formula_6 are in formula_4.
The Suslin operation on collections of subsets of formula_2 has the property that formula_17. The family formula_11 is closed under taking countable unions or intersections, but is not in general closed under taking complements.
If formula_4 is the family of closed subsets of a topological space, then the elements of formula_11 are called Suslin sets, or analytic sets if the space is a Polish space.
Example.
For each finite sequence formula_21, let formula_22 be the infinite sequences which extend formula_23. 
This is a clopen subset of formula_24.
If formula_2 is a Polish space and formula_26 is a continuous function, let formula_27.
Then formula_1 is a Suslin scheme consisting of closed subsets of formula_2 and formula_30.