Suslin surgery

In mathematics , the Suslin operation is a construction method of descriptive set theory with which numerous sets can be constructed.

Historically, it has its origin in the observation that one can construct all Borel sets from open or closed sets through complement formation and countable unions and averages , but that the images of Borel sets under orthogonal projections do not necessarily have to be Borel sets and one with the method proposed by Suslin contains a larger class of sets than just the Borel sets.

definition

Let be a family of sets indexed over the finite sequences of natural numbers . Applying the Suslin operation to this system of sets gives the set by definition ${\displaystyle \left\{E_{n_{1},\ldots ,n_{k}}\right\}}$ ${\displaystyle {n_{1},\ldots ,n_{k}}}$

${\displaystyle \bigcup _{n_{1},\ldots ,n_{k},\ldots }\;\bigcap _{k=1}^{\infty }E_{n_{1},\ldots ,n_{k}}}$,

where the union takes place over all infinite sequences . ${\displaystyle (n_{1},\ldots ,n_{k},\ldots )}$

The sets obtained from the family of open (or closed) subsets of a given space by the Suslin operation are called analytic sets .

literature

• PS Aleksandrov: Sur la puissance des ensembles measurables B , CR Acad. Sci. Paris 162 (1916), pp. 323-325
• M.Ya. Suslin: Sur un définition des ensembles measurables B sans nombres transfinis , CR Acad. Sci. Paris 164 (1917), pp. 88-91

Web links

• A-operation (Encyclopedia of Mathematics)