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

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

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

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

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

If $M$ is a family of subsets of a set $X$, then ${\mathcal {A}}(M)$ is the family of subsets of $X$ obtained by applying the Suslin operation ${\mathcal {A}}$ to all collections as above where all the sets $M_{n_{1},...,n_{k}}$ are in $M$. The Suslin operation on collections of subsets of $X$ has the property that ${\mathcal {A}}({\mathcal {A}}(M))={\mathcal {A}}(M)$. The family ${\mathcal {A}}(M)$ is closed under taking countable intersections and—if $X\in M$—countable unions, 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 ${\mathcal {A}}(M)$ are called Suslin sets, or analytic sets if the space is a Polish space.

Example

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

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