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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://www.encyclopediaofmath.org/index.php?title=A-set&oldid=34102
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article