# A-operation

operation ${\mathcal A}$

A set-theoretical operation, discovered by P.S. Aleksandrov [1] (see also [2], [3]). Let $\{ E _ {n _ {1} \dots n _ {k} } \}$ be a system of sets indexed by all finite sequences of natural numbers. The set

$$P = \cup _ {n _ {1} \dots n _ {k} , . . } \cap _ { k=1 } ^ \infty E _ {n _ {1} {} \dots n _ {k} } ,$$

where the union is over all infinite sequences of natural numbers, is called the result of the ${\mathcal A}$- operation applied to the system $\{ E _ {n _ {1} \dots n _ {k} } \}$.

The use of the ${\mathcal A}$- operation for the system of intervals of the number line gives sets (called ${\mathcal A}$- sets in honour of Aleksandrov) which need not be Borel sets (see ${\mathcal A}$- set; Descriptive set theory). The ${\mathcal A}$- operation is stronger than the operation of countable union and countable intersection, and is idempotent. With respect to ${\mathcal A}$- operations, the Baire property (of subsets of an arbitrary topological space) and the property of being Lebesgue measurable are invariant.

#### References

 [1] P.S. Aleksandrov, C.R. Acad. Sci. Paris , 162 (1916) pp. 323–325 [2] P.S. Aleksandrov, "Theory of functions of a real variable and the theory of topological spaces" , Moscow (1978) (In Russian) [3] A.N. Kolmogorov, "P.S. Aleksandrov and the theory of $\sigma \, \delta$-operations" Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 275–278 (In Russian) [4] M.Ya. Suslin, C.R. Acad. Sci. Paris , 164 (1917) pp. 88–91 [5] N.N. Luzin, , Collected works , 2 , Moscow (1958) pp. 284 (In Russian) [6] K. Kuratowski, "Topology" , 1–2 , Acad. Press (1966–1968) (Translated from French)

The ${\mathcal A}$- operation is in the West usually attributed to M.Ya. Suslin [4], and is therefore also called the Suslin operation, the Suslin ${\mathcal A}$- operation or the Suslin operation ${\mathcal A}$. ${\mathcal A}$- sets are usually called analytic sets.