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

$𝒜P=\bigcup _{x\in {\omega }^{\omega }}\bigcap _{n\in \omega }{P}_{x\upharpoonright n}$

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 $M_{n_1,\mathrm{...},n_k}$. The result of the Suslin operation is the set

$𝒜(M)=\bigcup (M_{n_1}\cap M_{n_1,n_2}\cap M_{n_1,n_2,n_3}\cap \dots )$

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 $𝒜(M)$ is the family of subsets of $X$ obtained by applying the Suslin operation $𝒜$ to all collections as above where all the sets $M_{n_1,\mathrm{...},n_k}$ are in $M$. The Suslin operation on collections of subsets of $X$ has the property that $𝒜(𝒜(M))=𝒜(M)$. The family $𝒜(M)$ 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 $𝒜(M)$ 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 $𝒜P=f[{\omega }^{\omega }]$.

References

• Aleksandrov, P. S. (1916), "Sur la puissance des ensembles measurables B", C. R. Acad. Sci. Paris, 162: 323–325
• Suslin, M. Ya. (1917), "Sur un définition des ensembles measurables B sans nombres transfinis", C. R. Acad. Sci. Paris, 164: 88–91