Namespaces
Variants
Actions

Difference between revisions of "Luzin sieve"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
An arbitrary mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611001.png" /> that puts each dyadic fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611002.png" /> into correspondence with a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611003.png" />. As a rule, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611004.png" /> is assumed to be a complete separable [[Metric space|metric space]]. It was introduced by N.N. Luzin [[#References|[1]]]. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611005.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611006.png" /> such that there is an infinite sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611007.png" /> that satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611008.png" /> is said to be sifted through the Luzin sieve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l0611009.png" />. For every [[A-operation|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110010.png" />-operation]] there is a Luzin sieve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110011.png" /> such that the result of this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110012.png" />-operation is sifted through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110013.png" />. The main result concerning the Luzin sieve is that a [[Luzin set|Luzin set]] of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110014.png" />-th class (or of the projective class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110015.png" />) is invariant under the operation of sifting through the Luzin sieve for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110016.png" />.
+
<!--
 +
l0611001.png
 +
$#A+1 = 25 n = 0
 +
$#C+1 = 25 : ~/encyclopedia/old_files/data/L061/L.0601100 Luzin sieve
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
An arbitrary mapping  $  W :  \mathbf Q _ {0} \rightarrow 2  ^ {X} $
 +
that puts each dyadic fraction $  r \in \mathbf Q _ {0} $
 +
into correspondence with a subset $  W _ {r} \subset  X $.  
 +
As a rule, $  X $
 +
is assumed to be a complete separable [[Metric space|metric space]]. It was introduced by N.N. Luzin [[#References|[1]]]. The set $  A $
 +
of points $  x \in X $
 +
such that there is an infinite sequence $  r _ {1} < r _ {2} < \dots $
 +
that satisfies the condition $  x \in W _ {r _ {1}  } \cap W _ {r _ {2}  } \cap \dots $
 +
is said to be sifted through the Luzin sieve $  W $.  
 +
For every [[A-operation| $  {\mathcal A} $-
 +
operation]] there is a Luzin sieve $  W $
 +
such that the result of this $  {\mathcal A} $-
 +
operation is sifted through $  W $.  
 +
The main result concerning the Luzin sieve is that a [[Luzin set|Luzin set]] of the $  n $-
 +
th class (or of the projective class $  L _ {n} $)  
 +
is invariant under the operation of sifting through the Luzin sieve for 0 \neq n \neq 2 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Luzin,  "Sur les ensembles analytiques"  ''Fund. Math.'' , '''10'''  (1927)  pp. 1–95</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Kuratowski,  "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Luzin,  "Sur les ensembles analytiques"  ''Fund. Math.'' , '''10'''  (1927)  pp. 1–95</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Kuratowski,  "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Line 11: Line 36:
 
This notion has nothing to do with the notion of sieve used by N. Bourbaki [[#References|[a1]]] while proving one of the Luzin theorems. A Bourbaki sieve is just a way to write a disjoint Suslin scheme.
 
This notion has nothing to do with the notion of sieve used by N. Bourbaki [[#References|[a1]]] while proving one of the Luzin theorems. A Bourbaki sieve is just a way to write a disjoint Suslin scheme.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110017.png" /> is an [[Analytic set|analytic set]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110018.png" /> is a Luzin sieve for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110019.png" /> consisting of closed sets, then, as one readily sees, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110021.png" />. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110023.png" />, are called the constituents of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110024.png" /> determined by the sieve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061100/l06110025.png" />.
+
If $  A $
 +
is an [[Analytic set|analytic set]] and $  W = \{ {W _ {r} } : {r \in \mathbf Q } \} $
 +
is a Luzin sieve for $  A $
 +
consisting of closed sets, then, as one readily sees, $  X \setminus  A = \{ {x } : {M _ {x}  \textrm{ is  well\AAh ordered  by  }  \geq  } \} $,  
 +
where $  M _ {x} = \{ {r } : {x \in W _ {r} } \} $.  
 +
The sets $  A _  \alpha  = \{ {x \in X \setminus  A } : {\textrm{ the  order  type  of  }  M _ {x } \textrm{ is }  \alpha } \} $,  
 +
where $  \alpha < \omega _ {1} $,  
 +
are called the constituents of the set $  X \setminus  A $
 +
determined by the sieve $  W $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  pp. Chapt. 10  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  pp. Chapt. 10  (Translated from French)</TD></TR></table>

Latest revision as of 04:11, 6 June 2020


An arbitrary mapping $ W : \mathbf Q _ {0} \rightarrow 2 ^ {X} $ that puts each dyadic fraction $ r \in \mathbf Q _ {0} $ into correspondence with a subset $ W _ {r} \subset X $. As a rule, $ X $ is assumed to be a complete separable metric space. It was introduced by N.N. Luzin [1]. The set $ A $ of points $ x \in X $ such that there is an infinite sequence $ r _ {1} < r _ {2} < \dots $ that satisfies the condition $ x \in W _ {r _ {1} } \cap W _ {r _ {2} } \cap \dots $ is said to be sifted through the Luzin sieve $ W $. For every $ {\mathcal A} $- operation there is a Luzin sieve $ W $ such that the result of this $ {\mathcal A} $- operation is sifted through $ W $. The main result concerning the Luzin sieve is that a Luzin set of the $ n $- th class (or of the projective class $ L _ {n} $) is invariant under the operation of sifting through the Luzin sieve for $ 0 \neq n \neq 2 $.

References

[1] N.N. Luzin, "Sur les ensembles analytiques" Fund. Math. , 10 (1927) pp. 1–95
[2] C. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)

Comments

A Luzin set in the sense of the article is invariably called a projective set in the West. The Luzin sieve has been an extremely powerful tool in descriptive set theory; it gave rise, with other techniques, to the modern use of countable ordinals in this theory. For more details and references see Descriptive set theory.

This notion has nothing to do with the notion of sieve used by N. Bourbaki [a1] while proving one of the Luzin theorems. A Bourbaki sieve is just a way to write a disjoint Suslin scheme.

If $ A $ is an analytic set and $ W = \{ {W _ {r} } : {r \in \mathbf Q } \} $ is a Luzin sieve for $ A $ consisting of closed sets, then, as one readily sees, $ X \setminus A = \{ {x } : {M _ {x} \textrm{ is well\AAh ordered by } \geq } \} $, where $ M _ {x} = \{ {r } : {x \in W _ {r} } \} $. The sets $ A _ \alpha = \{ {x \in X \setminus A } : {\textrm{ the order type of } M _ {x } \textrm{ is } \alpha } \} $, where $ \alpha < \omega _ {1} $, are called the constituents of the set $ X \setminus A $ determined by the sieve $ W $.

References

[a1] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) pp. Chapt. 10 (Translated from French)
How to Cite This Entry:
Luzin sieve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_sieve&oldid=12686
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article