Suslin operation

Summary

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).

DefinitionsEdit

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 set

 

Alternatively, 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 unions or intersections, 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.

ExampleEdit

For each finite sequence  , let   be the infinite sequences that extend  . This is a clopen subset of  . If   is a Polish space and   is a continuous function, let  . Then   is a Suslin scheme consisting of closed subsets of   and  .

ReferencesEdit

  • Aleksandrov, P. S. (1916), "Sur la puissance des ensembles measurables B", C. R. Acad. Sci. Paris, 162: 323–325
  • "A-operation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Suslin, M. Ya. (1917), "Sur un définition des ensembles measurables B sans nombres transfinis", C. R. Acad. Sci. Paris, 164: 88–91