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''analytic set, in a complete separable metric space''
 
''analytic set, in a complete separable metric space''
  
A continuous image of a Borel set. Since any Borel set is a continuous image of the set of irrational numbers, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a0100702.png" />-set can be defined as a continuous image of the set of irrational numbers. A countable intersection and a countable union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a0100703.png" />-sets is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a0100704.png" />-set. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a0100705.png" />-set is Lebesgue-measurable. The property of being an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a0100706.png" />-set is invariant relative to Borel-measurable mappings, and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a0100707.png" />-operations (cf. [[A-operation|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a0100708.png" />-operation]]). Moreover, for a set to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a0100709.png" />-set it is necessary and sufficient that it can be represented as the result of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007010.png" />-operation applied to a family of closed sets. There are examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007011.png" />-sets which are not Borel sets; thus, in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007012.png" /> of all closed subsets of the unit interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007013.png" /> of the real numbers, the set of all closed uncountable sets is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007014.png" />-set, but is not Borel. Any uncountable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007015.png" />-set topologically contains a perfect Cantor set. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007016.png" />-sets "realize" the continuum hypothesis: their cardinality is either finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007017.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007018.png" />. The [[Luzin separability principles|Luzin separability principles]] hold for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007019.png" />-sets.
+
A continuous image of a Borel set. Since any Borel set is a continuous
 +
image of the set of irrational numbers, an ${\mathcal A}$-set can be defined as a
 +
continuous image of the set of irrational numbers. A countable
 +
intersection and a countable union of ${\mathcal A}$-sets is an ${\mathcal A}$-set. Any
 +
${\mathcal A}$-set is Lebesgue-measurable. The property of being an ${\mathcal A}$-set is
 +
invariant relative to Borel-measurable mappings, and to ${\mathcal A}$-operations
 +
(cf.
 +
[[A-operation|${\mathcal A}$-operation]]). Moreover, for a set to be an ${\mathcal A}$-set
 +
it is necessary and sufficient that it can be represented as the
 +
result of an ${\mathcal A}$-operation applied to a family of closed sets. There
 +
are examples of ${\mathcal A}$-sets which are not Borel sets; thus, in the space
 +
$2^I$ of all closed subsets of the unit interval $I$ of the real
 +
numbers, the set of all closed uncountable sets is an ${\mathcal A}$-set, but is
 +
not Borel. Any uncountable ${\mathcal A}$-set topologically contains a perfect
 +
Cantor set. Thus, ${\mathcal A}$-sets "realize" the continuum hypothesis: their
 +
cardinality is either finite, $\aleph_0$ or $2^{\aleph_0}$. The
 +
[[Luzin separability principles|Luzin separability principles]] hold
 +
for ${\mathcal A}$-sets.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski,   "Topology" , '''1''' , Acad. Press (1966) (Translated from French)</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></table>
+
<table><TR><TD valign="top">[1]</TD> <TD
 +
valign="top"> K. Kuratowski, "Topology" , '''1''' , Acad. Press (1966)
 +
(Translated from French)</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></table>
  
  
  
 
====Comments====
 
====Comments====
Nowadays the class of analytic sets is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007020.png" />, and the class of co-analytic sets (cf. [[CA-set|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007021.png" />-set]]) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010070/a01007022.png" />.
+
Nowadays the class of analytic sets is denoted by
 +
$\Sigma_1^1$, and the class of co-analytic sets (cf.
 +
[[CA-set|${\mathcal CA}$-set]]) by $\Pi_1^1$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> T.J. Jech,   "The axiom of choice" , North-Holland (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Y.N. Moschovakis,   "Descriptive set theory" , North-Holland (1980)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD
 +
valign="top"> T.J. Jech, "The axiom of choice" , North-Holland
 +
(1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
 +
Y.N. Moschovakis, "Descriptive set theory" , North-Holland
 +
(1980)</TD></TR></table>

Revision as of 21:54, 11 September 2011

analytic set, in a complete separable metric space

A continuous image of a Borel set. Since any Borel set is a continuous image of the set of irrational numbers, an ${\mathcal A}$-set can be defined as a continuous image of the set of irrational numbers. A countable intersection and a countable union of ${\mathcal A}$-sets is an ${\mathcal A}$-set. Any ${\mathcal A}$-set is Lebesgue-measurable. The property of being an ${\mathcal A}$-set is invariant relative to Borel-measurable mappings, and to ${\mathcal A}$-operations (cf. ${\mathcal A}$-operation). Moreover, for a set to be an ${\mathcal A}$-set it is necessary and sufficient that it can be represented as the result of an ${\mathcal A}$-operation applied to a family of closed sets. There are examples of ${\mathcal A}$-sets which are not Borel sets; thus, in the space $2^I$ of all closed subsets of the unit interval $I$ of the real numbers, the set of all closed uncountable sets is an ${\mathcal A}$-set, but is not Borel. Any uncountable ${\mathcal A}$-set topologically contains a perfect Cantor set. Thus, ${\mathcal A}$-sets "realize" the continuum hypothesis: their cardinality is either finite, $\aleph_0$ or $2^{\aleph_0}$. The Luzin separability principles hold for ${\mathcal A}$-sets.

References

[1] K. Kuratowski, "Topology" , 1 , Acad. Press (1966) (Translated from French)
[2] N.N. [N.N. Luzin] Lusin, "Leçons sur les ensembles

analytiques et leurs applications" , Gauthier-Villars

(1930)


Comments

Nowadays the class of analytic sets is denoted by $\Sigma_1^1$, and the class of co-analytic sets (cf. ${\mathcal CA}$-set) by $\Pi_1^1$.

References

[a1] T.J. Jech, "The axiom of choice" , North-Holland (1973)
[a2]

Y.N. Moschovakis, "Descriptive set theory" , North-Holland

(1980)
How to Cite This Entry:
A-set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=A-set&oldid=19565
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article