Suslin operation explained

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 (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 l A P = c up _{x \in {\omega ^{\omega}} } \bigcap_ P_

Alternatively, suppose we have a Suslin scheme, in other words a function

M from finite sequences of positive integers n _{1,...,} n _{k} to sets M _{n _{1,...,} n _{k}} . The result of the Suslin operation is the set l A(M) = c up \left(M _{n _{1}} \cap M _{n _{1,} n _{2}} \cap M _{n _{1,} n _{2,} n _{3}} \cap ... \right) where the union is taken over all infinite sequences n _{1,...,} n _{k,} ...

If

M is a family of subsets of a set X , then l A(M) is the family of subsets of X obtained by applying the Suslin operation l 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 l A( l A(M)) = l A(M) . The family l 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 l 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 l AP = f[\omega ^{\omega} ]