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====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010020/a01002013.png" />-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>
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<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====
 
====Comments====

Latest revision as of 16:59, 1 July 2020


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=45193
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