Suslin operation
In mathematics, the Suslin operation ? 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 ?.
Definitions
A Suslin scheme is a family of subsets of a set indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the setAlternatively, suppose we have a Suslin scheme, in other words a function from finite sequences of positive integers to sets. The result of the Suslin operation is the set
where the union is taken over all infinite sequences
If is a family of subsets of a set, then is the family of subsets of obtained by applying the Suslin operation to all collections as above where all the sets are in.
The Suslin operation on collections of subsets of has the property that. The family is closed under taking countable intersections and—if —countable unions, but is not in general closed under taking complements.
If is the family of closed subsets of a topological space, then the elements of are called Suslin sets, or analytic sets if the space is a Polish space.
Example
For each finite sequence, let be the infinite sequences that extend.This is a clopen subset of Baire space |.
If is a Polish space and is a continuous function, let.
Then is a Suslin scheme consisting of closed subsets of and.