View source for A-set

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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.
[[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.

====Comments====
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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''1''' , Acad. Press (1966) (Translated from French) {{ZBL|0158.40802}}</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">[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>

[[Category:Topology]]