BREAKING NEWS

## 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 $P=\{P_{s}:s\in \omega ^{<\omega }\}$  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 unions or intersections, 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 ${\mathcal {A}}P=f[\omega ^{\omega }]$ .