# 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) |

#### Comments

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.

**How to Cite This Entry:**

A-operation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=A-operation&oldid=50301