Suslin operation
In mathematics, the Suslin operation π 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 π (a calligraphic capital letter A).
Definitions
A Suslin scheme is a family P = {Ps: s β Ο>Ο} of subsets of a set X indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set πP = UxβΟΟ β©nβΟ PxβΎn.
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
- π(M) = βͺ (Mn1 β© Mn1,n2 β© Mn1,n2, n3 β© ...)
where the union is taken over all infinite sequences n1,...,nk,...
If M is a family of subsets of a set X, then π(M) is the family of subsets of X obtained by applying the Suslin operation π 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 π(π(M)) = π(M). The family π(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 π(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: 323β325
- Hazewinkel, Michiel, ed. (2001), "A-operation", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Suslin, M. Ya. (1917), "Sur un dΓ©finition des ensembles measurables B sans nombres transfinis", C.R. Acad. Sci. Paris, 164: 88β91