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[[Luzin separability principles|Luzin separability principles]] hold
 
[[Luzin separability principles|Luzin separability principles]] hold
 
for ${\mathcal A}$-sets.
 
for ${\mathcal A}$-sets.
 
====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>
 
 
 
  
 
====Comments====
 
====Comments====
Line 37: Line 27:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD
+
<table>
valign="top"> T.J. Jech, "The axiom of choice" , North-Holland
+
<TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''1''' , Acad. Press (1966) (Translated from French)</TD></TR>
(1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">
+
<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>
Y.N. Moschovakis, "Descriptive set theory" , North-Holland
+
<TR><TD valign="top">[a1]</TD> <TD valign="top"> T.J. Jech, "The axiom of choice" , North-Holland (1973)</TD></TR>
(1980)</TD></TR></table>
+
<TR><TD valign="top">[a2]</TD> <TD valign="top"> Y.N. Moschovakis, "Descriptive set theory" , North-Holland (1980)</TD></TR>
 +
</table>
  
 
[[Category:Topology]]
 
[[Category:Topology]]

Revision as of 19:34, 25 March 2023

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.

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

[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)
[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=53232
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article