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 [math]\displaystyle{ P = \{ P_s: s \in \omega^{\lt \omega} \} }[/math] of subsets of a set [math]\displaystyle{ X }[/math] indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set

[math]\displaystyle{ \mathcal A P = \bigcup_{x \in {\omega ^ \omega}} \bigcap_{n \in \omega} P_{x \upharpoonright n} }[/math]

Alternatively, suppose we have a Suslin scheme, in other words a function [math]\displaystyle{ M }[/math] from finite sequences of positive integers [math]\displaystyle{ n_1,\dots, n_k }[/math] to sets [math]\displaystyle{ M_{n_1,..., n_k} }[/math]. The result of the Suslin operation is the set

[math]\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) }[/math]

where the union is taken over all infinite sequences [math]\displaystyle{ n_1,\dots, n_k, \dots }[/math]

If [math]\displaystyle{ M }[/math] is a family of subsets of a set [math]\displaystyle{ X }[/math], then [math]\displaystyle{ \mathcal A(M) }[/math] is the family of subsets of [math]\displaystyle{ X }[/math] obtained by applying the Suslin operation [math]\displaystyle{ \mathcal A }[/math] to all collections as above where all the sets [math]\displaystyle{ M_{n_1,..., n_k} }[/math] are in [math]\displaystyle{ M }[/math]. The Suslin operation on collections of subsets of [math]\displaystyle{ X }[/math] has the property that [math]\displaystyle{ \mathcal A(\mathcal A(M)) = \mathcal A(M) }[/math]. The family [math]\displaystyle{ \mathcal A(M) }[/math] is closed under taking countable unions or intersections, but is not in general closed under taking complements.

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

Example

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

References

• Aleksandrov, P. S. (1916), "Sur la puissance des ensembles measurables B", C. R. Acad. Sci. Paris 162: 323–325 
• Hazewinkel, Michiel, ed. (2001), "A-operation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=A-operation 
• Suslin, M. Ya. (1917), "Sur un définition des ensembles measurables B sans nombres transfinis", C. R. Acad. Sci. Paris 164: 88–91