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''projective set''
 
''projective set''
  
A subset of a complete separable metric space, defined by induction as follows. The Luzin sets of class 0 are the Borel sets (cf. [[Borel set|Borel set]]). The Luzin sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l0610901.png" /> are continuous images of Luzin sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l0610902.png" />. The Luzin sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l0610903.png" /> are complements of Luzin sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l0610904.png" />. In particular, Luzin sets of class 1, that is, continuous images of Borel sets, are called analytic sets, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l0610906.png" />-sets or Suslin sets (cf. [[A-set|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l0610907.png" />-set]]; [[Analytic set|Analytic set]]). The concept of a Luzin set is due to N.N. Luzin [[#References|[1]]]. If the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l0610908.png" /> are Luzin sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l0610909.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109011.png" /> are also Luzin sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109012.png" />. If the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109013.png" /> are Luzin sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109014.png" /> lying in complete separable metric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109015.png" />, then the direct product (finite or countable) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109016.png" /> is a Luzin set of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109017.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109018.png" />. A Luzin set of odd class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109019.png" /> situated in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109020.png" /> coincides with the projection of a set of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109021.png" /> situated in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109022.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109023.png" /> of irrational numbers in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109024.png" /> contains, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109025.png" />, a Luzin set of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109026.png" /> that is not a Luzin set of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109027.png" />; the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109028.png" /> also contains sets that are not Luzin sets.
+
A subset of a complete separable metric space, defined by induction as follows. The Luzin sets of class 0 are the Borel sets (cf. [[Borel set|Borel set]]). The Luzin sets of class $  2 n + 1 $
 +
are continuous images of Luzin sets of class $  2 n $.  
 +
The Luzin sets of class $  2 n $
 +
are complements of Luzin sets of class $  2 n - 1 $.  
 +
In particular, Luzin sets of class 1, that is, continuous images of Borel sets, are called analytic sets, or $  {\mathcal A} $-
 +
sets or Suslin sets (cf. [[A-set| $  {\mathcal A} $-
 +
set]]; [[Analytic set|Analytic set]]). The concept of a Luzin set is due to N.N. Luzin [[#References|[1]]]. If the sets $  P _ {i} $
 +
are Luzin sets of class $  n $,  
 +
then $  \cup _ {i=} 1  ^ {k} P _ {i} $
 +
and $  \cap _ {i=} 1  ^ {k} P _ {i} $
 +
are also Luzin sets of class $  n $.  
 +
If the sets $  P _ {i} \subset  X _ {i} $
 +
are Luzin sets of class $  n $
 +
lying in complete separable metric spaces $  X _ {i} $,  
 +
then the direct product (finite or countable) $  \prod _ {i} P _ {i} $
 +
is a Luzin set of class $  n $
 +
in the space $  \prod _ {i} X _ {i} $.  
 +
A Luzin set of odd class $  n $
 +
situated in a space $  X $
 +
coincides with the projection of a set of class $  n- 1 $
 +
situated in $  X \times X $.  
 +
The space $  X $
 +
of irrational numbers in the interval $  [ 0 , 1 ] $
 +
contains, for any $  n> 0 $,  
 +
a Luzin set of class $  n $
 +
that is not a Luzin set of class < n $;  
 +
the space $  X $
 +
also contains sets that are not Luzin sets.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin,  "Sur un problème de M. Emile Borel et les ensemble projectifs de M. Henri Lebesgue"  ''C.R. Acad. Sci. Paris'' , '''180'''  (1925)  pp. 1318–1320</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin,  "Leçons sur les ensembles analytiques et leurs applications" , Gauthier-Villars  (1930)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. 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. [N.N. Luzin] Lusin,  "Sur un problème de M. Emile Borel et les ensemble projectifs de M. Henri Lebesgue"  ''C.R. Acad. Sci. Paris'' , '''180'''  (1925)  pp. 1318–1320</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. [N.N. Luzin] Lusin,  "Leçons sur les ensembles analytiques et leurs applications" , Gauthier-Villars  (1930)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The term  "Luzin set"  is in the West almost exclusively used for a subset of the real line whose intersection with every nowhere-dense set is countable (see [[Luzin space|Luzin space]]). The sets discussed in the main article above are almost exclusively called projective sets (cf. [[Projective set|Projective set]]). The sets of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109029.png" /> are generally called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109030.png" />-sets and those of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109031.png" /> are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061090/l06109032.png" />-sets. See [[Descriptive set theory|Descriptive set theory]].
+
The term  "Luzin set"  is in the West almost exclusively used for a subset of the real line whose intersection with every nowhere-dense set is countable (see [[Luzin space|Luzin space]]). The sets discussed in the main article above are almost exclusively called projective sets (cf. [[Projective set|Projective set]]). The sets of class $  2 n + 1 $
 +
are generally called $  \Sigma _ {n}  ^ {1} $-
 +
sets and those of class $  2 n $
 +
are called $  \Pi _ {n}  ^ {1} $-
 +
sets. See [[Descriptive set theory|Descriptive set theory]].
  
 
All important problems about projective sets have received satisfactory answers during the last three decades, see [[Descriptive set theory|Descriptive set theory]] and [[Luzin problem|Luzin problem]].
 
All important problems about projective sets have received satisfactory answers during the last three decades, see [[Descriptive set theory|Descriptive set theory]] and [[Luzin problem|Luzin problem]].

Latest revision as of 04:11, 6 June 2020


projective set

A subset of a complete separable metric space, defined by induction as follows. The Luzin sets of class 0 are the Borel sets (cf. Borel set). The Luzin sets of class $ 2 n + 1 $ are continuous images of Luzin sets of class $ 2 n $. The Luzin sets of class $ 2 n $ are complements of Luzin sets of class $ 2 n - 1 $. In particular, Luzin sets of class 1, that is, continuous images of Borel sets, are called analytic sets, or $ {\mathcal A} $- sets or Suslin sets (cf. $ {\mathcal A} $- set; Analytic set). The concept of a Luzin set is due to N.N. Luzin [1]. If the sets $ P _ {i} $ are Luzin sets of class $ n $, then $ \cup _ {i=} 1 ^ {k} P _ {i} $ and $ \cap _ {i=} 1 ^ {k} P _ {i} $ are also Luzin sets of class $ n $. If the sets $ P _ {i} \subset X _ {i} $ are Luzin sets of class $ n $ lying in complete separable metric spaces $ X _ {i} $, then the direct product (finite or countable) $ \prod _ {i} P _ {i} $ is a Luzin set of class $ n $ in the space $ \prod _ {i} X _ {i} $. A Luzin set of odd class $ n $ situated in a space $ X $ coincides with the projection of a set of class $ n- 1 $ situated in $ X \times X $. The space $ X $ of irrational numbers in the interval $ [ 0 , 1 ] $ contains, for any $ n> 0 $, a Luzin set of class $ n $ that is not a Luzin set of class $ < n $; the space $ X $ also contains sets that are not Luzin sets.

References

[1] N.N. [N.N. Luzin] Lusin, "Sur un problème de M. Emile Borel et les ensemble projectifs de M. Henri Lebesgue" C.R. Acad. Sci. Paris , 180 (1925) pp. 1318–1320
[2] N.N. [N.N. Luzin] Lusin, "Leçons sur les ensembles analytiques et leurs applications" , Gauthier-Villars (1930)
[3] K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)

Comments

The term "Luzin set" is in the West almost exclusively used for a subset of the real line whose intersection with every nowhere-dense set is countable (see Luzin space). The sets discussed in the main article above are almost exclusively called projective sets (cf. Projective set). The sets of class $ 2 n + 1 $ are generally called $ \Sigma _ {n} ^ {1} $- sets and those of class $ 2 n $ are called $ \Pi _ {n} ^ {1} $- sets. See Descriptive set theory.

All important problems about projective sets have received satisfactory answers during the last three decades, see Descriptive set theory and Luzin problem.

References

[a1] Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)
[a2] T.J. Jech, "Set theory" , Acad. Press (1978) pp. 523ff (Translated from German)
How to Cite This Entry:
Luzin set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_set&oldid=47723
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article