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The complement of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c0200303.png" />-set in a complete separable metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c0200304.png" />; that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c0200305.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c0200306.png" />-set if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c0200307.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c0200308.png" />-set, or, in other words a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c0200309.png" />-set is a [[Projective set|projective set]] of class 2. There is an example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003010.png" />-set that is not an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003011.png" />-set. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003012.png" />-set is a one-to-one continuous image of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003013.png" />-set (Mazurkiewicz's theorem).
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A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003014.png" /> is called a value of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003016.png" /> of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003017.png" /> if there is one and only one point such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003018.png" />. The values of order 1 of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003019.png" />-measurable mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003020.png" /> on an arbitrary Borel set form a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003021.png" />-set (Luzin's theorem). The converse is true: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003023.png" /> be any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003024.png" />-set belonging to a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003025.png" />. Then there is a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003026.png" /> defined on a closed subset of the irrational numbers such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003027.png" /> is the set of points of order 1 of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003028.png" />. Kuratowski's reduction theorem: Given an infinite sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003029.png" />-sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003030.png" /> there is a sequence of disjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003031.png" />-sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003034.png" />.
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The complement of an  $  {\mathcal A} $-
 +
set in a complete separable metric space  $  X $;
 +
that is,  $  P \subset  X $
 +
is a  $  C {\mathcal A} $-
 +
set if  $  X \setminus  P $
 +
is an  $  {\mathcal A} $-
 +
set, or, in other words a  $  C {\mathcal A} $-
 +
set is a [[Projective set|projective set]] of class 2. There is an example of a  $  C {\mathcal A} $-
 +
set that is not an  $  {\mathcal A} $-
 +
set. Any  $  {\mathcal A} $-
 +
set is a one-to-one continuous image of some  $  C {\mathcal A} $-
 +
set (Mazurkiewicz's theorem).
 +
 
 +
A point  $  y $
 +
is called a value of order $  1 $
 +
of a mapping $  f $
 +
if there is one and only one point such that $  y = f (x) $.  
 +
The values of order 1 of a $  B $-
 +
measurable mapping $  f $
 +
on an arbitrary Borel set form a $  C {\mathcal A} $-
 +
set (Luzin's theorem). The converse is true: Let $  C $
 +
be any $  C {\mathcal A} $-
 +
set belonging to a space $  X $.  
 +
Then there is a continuous function $  f $
 +
defined on a closed subset of the irrational numbers such that $  C $
 +
is the set of points of order 1 of $  f $.  
 +
Kuratowski's reduction theorem: Given an infinite sequence of $  C {\mathcal A} $-
 +
sets $  U  ^ {1} , U  ^ {2} \dots $
 +
there is a sequence of disjoint $  C {\mathcal A} $-
 +
sets $  V  ^ {1} , V  ^ {2} \dots $
 +
such that $  V  ^ {n} \subset  U  ^ {n} $
 +
and $  \cup _ {n=1}  ^  \infty  V  ^ {n} = \cup _ {n=1}  ^  \infty  U  ^ {n} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">  K. Kuratowski,  "Topology" , '''1''' , Acad. Press  (1966)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003035.png" />-set is also called a co-analytic set, their class is nowadays denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003036.png" />. See also [[A-set|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020030/c02003037.png" />-set]].
+
A $  C {\mathcal A} $-
 +
set is also called a co-analytic set, their class is nowadays denoted by $  \Pi _ {1}  ^ {1} $.  
 +
See also [[A-set| $  {\mathcal A} $-
 +
set]].

Latest revision as of 06:29, 30 May 2020


The complement of an $ {\mathcal A} $- set in a complete separable metric space $ X $; that is, $ P \subset X $ is a $ C {\mathcal A} $- set if $ X \setminus P $ is an $ {\mathcal A} $- set, or, in other words a $ C {\mathcal A} $- set is a projective set of class 2. There is an example of a $ C {\mathcal A} $- set that is not an $ {\mathcal A} $- set. Any $ {\mathcal A} $- set is a one-to-one continuous image of some $ C {\mathcal A} $- set (Mazurkiewicz's theorem).

A point $ y $ is called a value of order $ 1 $ of a mapping $ f $ if there is one and only one point such that $ y = f (x) $. The values of order 1 of a $ B $- measurable mapping $ f $ on an arbitrary Borel set form a $ C {\mathcal A} $- set (Luzin's theorem). The converse is true: Let $ C $ be any $ C {\mathcal A} $- set belonging to a space $ X $. Then there is a continuous function $ f $ defined on a closed subset of the irrational numbers such that $ C $ is the set of points of order 1 of $ f $. Kuratowski's reduction theorem: Given an infinite sequence of $ C {\mathcal A} $- sets $ U ^ {1} , U ^ {2} \dots $ there is a sequence of disjoint $ C {\mathcal A} $- sets $ V ^ {1} , V ^ {2} \dots $ such that $ V ^ {n} \subset U ^ {n} $ and $ \cup _ {n=1} ^ \infty V ^ {n} = \cup _ {n=1} ^ \infty U ^ {n} $.

References

[1] K. Kuratowski, "Topology" , 1 , Acad. Press (1966) (Translated from French)

Comments

A $ C {\mathcal A} $- set is also called a co-analytic set, their class is nowadays denoted by $ \Pi _ {1} ^ {1} $. See also $ {\mathcal A} $- set.

How to Cite This Entry:
CA-set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CA-set&oldid=46182
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article