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A  "countably-ramified"  system of sets, i.e. a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a0100802.png" /> of subsets of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a0100803.png" />, indexed by all finite sequences of natural numbers. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a0100804.png" />-system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a0100805.png" /> is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a0100807.png" />. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a0100808.png" /> of elements of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a0100809.png" />-system indexed by all segments of one and the same finite sequence of natural numbers is called a chain of this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008011.png" />-system. The intersection of all elements of a chain is called its kernel, and the union of all kernels of all chains of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008013.png" />-system is called the kernel of this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008014.png" />-system, or the result of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008016.png" />-operation applied to this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008017.png" />-system, or the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008020.png" />-set generated by this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008021.png" />-system. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008022.png" />-system can be regularized without changing the kernel (it suffices to put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008023.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008024.png" /> is a system of sets, then the kernels of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008025.png" />-system composed from the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008026.png" /> are called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008030.png" />-sets generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008031.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008032.png" />-sets generated by the closed sets of a topological space are called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008033.png" />-sets of this space.
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A  "countably-ramified"  system of sets, i.e. a family $  \{ A _ {n _ {1}  \dots n _ {k} } \} $
 +
of subsets of a set $  X $,  
 +
indexed by all finite sequences of natural numbers. An $  A $-
 +
system $  \{ A _ {n _ {1}  \dots n _ {k} } \} $
 +
is called regular if $  A _ {n _ {1}  \dots n _ {k} n _ {k+1} } \subset  A _ {n _ {1}  \dots n _ {k} } $.  
 +
A sequence $  A _ {n _ {1}  } \dots A _ {n _ {1}  \dots n _ {k} } \dots $
 +
of elements of an $  A $-
 +
system indexed by all segments of one and the same finite sequence of natural numbers is called a chain of this $  A $-
 +
system. The intersection of all elements of a chain is called its kernel, and the union of all kernels of all chains of an $  A $-
 +
system is called the kernel of this $  A $-
 +
system, or the result of the $  {\mathcal A} $-
 +
operation applied to this $  A $-
 +
system, or the $  {\mathcal A} $-
 +
set generated by this $  A $-
 +
system. Every $  A $-
 +
system can be regularized without changing the kernel (it suffices to put $  A _ {n _ {1}  \dots n _ {k} }  ^  \prime  = A _ {1} \cap \dots \cap A _ {n _ {1}  \dots n _ {k} } $).  
 +
If $  {\mathcal M} $
 +
is a system of sets, then the kernels of the $  A $-
 +
system composed from the elements of $  {\mathcal M} $
 +
are called the $  {\mathcal A} $-
 +
sets generated by $  {\mathcal M} $.  
 +
The $  {\mathcal A} $-
 +
sets generated by the closed sets of a topological space are called the $  {\mathcal A} $-
 +
sets of this space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , Acad. Press  (1966)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , Acad. Press  (1966)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008034.png" />-operation is an important tool in descriptive set theory. It was introduced by M.Ya. Suslin, hence it is also known as the Suslin operation (also Souslin operation). In this connection, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008035.png" />-system is also called a Suslin (Souslin) scheme. See also [[A-operation|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008036.png" />-operation]]; [[A-set|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010080/a01008037.png" />-set]].
+
The $  {\mathcal A} $-
 +
operation is an important tool in descriptive set theory. It was introduced by M.Ya. Suslin, hence it is also known as the Suslin operation (also Souslin operation). In this connection, an $  A $-
 +
system is also called a Suslin (Souslin) scheme. See also [[A-operation| $  {\mathcal A} $-
 +
operation]]; [[A-set| $  {\mathcal A} $-
 +
set]].
  
 
While [[#References|[2]]] is the standard reference for classical results, a modern approach can be found in [[#References|[a1]]].
 
While [[#References|[2]]] is the standard reference for classical results, a modern approach can be found in [[#References|[a1]]].

Latest revision as of 18:48, 5 April 2020


A "countably-ramified" system of sets, i.e. a family $ \{ A _ {n _ {1} \dots n _ {k} } \} $ of subsets of a set $ X $, indexed by all finite sequences of natural numbers. An $ A $- system $ \{ A _ {n _ {1} \dots n _ {k} } \} $ is called regular if $ A _ {n _ {1} \dots n _ {k} n _ {k+1} } \subset A _ {n _ {1} \dots n _ {k} } $. A sequence $ A _ {n _ {1} } \dots A _ {n _ {1} \dots n _ {k} } \dots $ of elements of an $ A $- system indexed by all segments of one and the same finite sequence of natural numbers is called a chain of this $ A $- system. The intersection of all elements of a chain is called its kernel, and the union of all kernels of all chains of an $ A $- system is called the kernel of this $ A $- system, or the result of the $ {\mathcal A} $- operation applied to this $ A $- system, or the $ {\mathcal A} $- set generated by this $ A $- system. Every $ A $- system can be regularized without changing the kernel (it suffices to put $ A _ {n _ {1} \dots n _ {k} } ^ \prime = A _ {1} \cap \dots \cap A _ {n _ {1} \dots n _ {k} } $). If $ {\mathcal M} $ is a system of sets, then the kernels of the $ A $- system composed from the elements of $ {\mathcal M} $ are called the $ {\mathcal A} $- sets generated by $ {\mathcal M} $. The $ {\mathcal A} $- sets generated by the closed sets of a topological space are called the $ {\mathcal A} $- sets of this space.

References

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2] K. Kuratowski, "Topology" , 1 , Acad. Press (1966) (Translated from French)

Comments

The $ {\mathcal A} $- operation is an important tool in descriptive set theory. It was introduced by M.Ya. Suslin, hence it is also known as the Suslin operation (also Souslin operation). In this connection, an $ A $- system is also called a Suslin (Souslin) scheme. See also $ {\mathcal A} $- operation; $ {\mathcal A} $- set.

While [2] is the standard reference for classical results, a modern approach can be found in [a1].

References

[a1] J.P.R. Christensen, "Topology and Borel structure" , North-Holland (1974)
How to Cite This Entry:
A-system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A-system&oldid=45245
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