# Analytic set

A subset of a complete separable metric space that is a continuous image of the space of irrational numbers. The concept of an analytic set was introduced by N.N. Luzin . His classical definition has been generalized to general metric and topological spaces.

1) An analytic set in an arbitrary topological space $X$ is a subset of that space that is the image of a closed subset of the space of irrational numbers under an upper semi-continuous multi-valued mapping with compact images of points and a closed graph . If $X$ is a Hausdorff space, the last-mentioned condition is automatically satisfied. If $X$ is metrizable, this definition is equivalent to the classical one.

2) In a complete separable metric space the classical analytic sets are identical with ${\mathcal A}$- sets (cf. ${\mathcal A}$- set). This fact forms the base of another definition of analytic sets (in their capacity as ${\mathcal A}$- sets) in general metric and topological spaces , , . In the class of completely-regular spaces, analytic sets in the sense of 1) are absolute analytic sets in the sense of 2). In the class of non-separable metrizable spaces definition 2) is employed, since definition 1) yields separable analytic sets.

3) An analytic set in a Hausdorff space , , is a continuous image of a subset of a compact space of type $F _ {\sigma \delta }$.

4) An analytic set is a continuous image of a set belonging to the family $K _ {\sigma \delta }$, where $K$ is the family of all closed compact subsets of some topological space . Definitions 1), 3) and 4) equivalent in the class of Hausdorff spaces.

5) For a generalization in another direction see ; $k$- analytic sets are obtained from closed sets of a topological space by generalized ${\mathcal A}$- operations (cf. ${\mathcal A}$- operation; the Baire space of countable weight is replaced by a Baire space of weight $k$) and are a generalization of analytic sets in the sense of 2).

How to Cite This Entry:
Analytic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_set&oldid=45180
This article was adapted from an original article by A.G. El'kin, E.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article