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''operation  $  {\mathcal A} $''

A set-theoretical operation, discovered by P.S. Aleksandrov [[#References|[1]]] (see also [[#References|[2]]], [[#References|[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 [[A-set| $  {\mathcal A} $-
set]]; [[Descriptive set theory|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|Baire property]] (of subsets of an arbitrary topological space) and the property of being Lebesgue measurable are invariant.

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

====Comments====
The  $  {\mathcal A} $-
operation is in the West usually attributed to M.Ya. Suslin [[#References|[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.