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 Alexandrov (1916) and Suslin (1917). 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 = {P_s: s ∈ ^ω>ω} of subsets of a set X indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set 𝓐P = U_x∈^ω_ω ∩_n∈ω P_x↾n.

Alternatively, suppose we have Suslin scheme, in other words a function M from finite sequences of positive integers n₁,...,n_k to sets M_n1,...,nk. The result of the Suslin operation is the set

𝓐(M) = ∪ (M_n1 ∩ M_n1,n2 ∩ M_{n1,n2, n3} ∩ ...)

where the union is taken over all infinite sequences n₁,...,n_k,...

If M is a family of subsets of a set X, then 𝓐(M) is the family of subsets of X obtained by applying the Suslin operation 𝓐 to all collections as above where all the sets M_n1,...,nk are in M. The Suslin operation on collections of subsets of X has the property that 𝓐(𝓐(M)) = 𝓐(M). The family 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 𝓐(M) are called Suslin sets, or analytic sets if the space is a Polish space.

References

• Aleksandrov, P. S. (1916), "Sur la puissance des ensembles measurables B", C.R. Acad. Sci. Paris 162: 323–325
• Hazewinkel, Michiel, ed. (2001), "A-operation", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Suslin, M. Ya. (1917), "Sur un définition des ensembles measurables B sans nombres transfinis", C.R. Acad. Sci. Paris 164: 88–91