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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).

Definitions

A Suslin scheme is a family ${\displaystyle P=\{P_{s}:s\in \omega ^{<\omega }\}}$  of subsets of a set ${\displaystyle X}$  indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set

${\displaystyle {\mathcal {A}}P=\bigcup _{x\in {\omega ^{\omega }}}\bigcap _{n\in \omega }P_{x\upharpoonright n}}$

Alternatively, suppose we have a Suslin scheme, in other words a function ${\displaystyle M}$  from finite sequences of positive integers ${\displaystyle n_{1},\dots ,n_{k}}$  to sets ${\displaystyle M_{n_{1},...,n_{k}}}$ . The result of the Suslin operation is the set

${\displaystyle {\mathcal {A}}(M)=\bigcup \left(M_{n_{1}}\cap M_{n_{1},n_{2}}\cap M_{n_{1},n_{2},n_{3}}\cap \dots \right)}$

where the union is taken over all infinite sequences ${\displaystyle n_{1},\dots ,n_{k},\dots }$

If ${\displaystyle M}$  is a family of subsets of a set ${\displaystyle X}$ , then ${\displaystyle {\mathcal {A}}(M)}$  is the family of subsets of ${\displaystyle X}$  obtained by applying the Suslin operation ${\displaystyle {\mathcal {A}}}$  to all collections as above where all the sets ${\displaystyle M_{n_{1},...,n_{k}}}$  are in ${\displaystyle M}$ . The Suslin operation on collections of subsets of ${\displaystyle X}$  has the property that ${\displaystyle {\mathcal {A}}({\mathcal {A}}(M))={\mathcal {A}}(M)}$ . The family ${\displaystyle {\mathcal {A}}(M)}$  is closed under taking countable unions or intersections, but is not in general closed under taking complements.

If ${\displaystyle M}$  is the family of closed subsets of a topological space, then the elements of ${\displaystyle {\mathcal {A}}(M)}$  are called Suslin sets, or analytic sets if the space is a Polish space.

Example

For each finite sequence ${\displaystyle s\in \omega ^{n}}$ , let ${\displaystyle N_{s}=\{x\in \omega ^{\omega }:x\upharpoonright n=s\}}$  be the infinite sequences that extend ${\displaystyle s}$ . This is a clopen subset of ${\displaystyle \omega ^{\omega }}$ . If ${\displaystyle X}$  is a Polish space and ${\displaystyle f:\omega ^{\omega }\to X}$  is a continuous function, let ${\displaystyle P_{s}={\overline {f[N_{s}]}}}$ . Then ${\displaystyle P=\{P_{s}:s\in \omega ^{<\omega }\}}$  is a Suslin scheme consisting of closed subsets of ${\displaystyle X}$  and ${\displaystyle {\mathcal {A}}P=f[\omega ^{\omega }]}$ .

References

• 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