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).

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 intersections and—if ${\displaystyle X\in M}$—countable unions, 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.

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 ${\displaystyle {\mathcal {A}}P=f[\omega ^{\omega }]}$.