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A set-theoretical operation, discovered by P.S. Aleksandrov [1] (see also [2], [3]). Let be a system of sets indexed by all finite sequences of natural numbers. The set

where the union is over all infinite sequences of natural numbers, is called the result of the -operation applied to the system .

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


[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 -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 -operation is in the West usually attributed to M.Ya. Suslin [4], and is therefore also called the Suslin operation, the Suslin -operation or the Suslin operation . -sets are usually called analytic sets.

How to Cite This Entry:
A-operation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.G. El'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article