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 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 ?P = Ux∈ωω ∩n∈ω Px↾n.
Alternatively, suppose we have Suslin scheme, in other words a function M from finite sequences of positive integers n1,...,nk to sets Mn1,...,nk. The result of the Suslin operation is the set
where the union is taken over all infinite sequences n1,...,nk,...
If M is a family of subsets of a set X, then ? is the family of subsets of X obtained by applying the Suslin operation ? to all collections as above where all the sets Mn1,...,nk are in M.
The Suslin operation on collections of subsets of X has the property that ? = ?. The family ? 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 ? are called Suslin sets, or analytic sets if the space is a Polish space.