# Analytic set

1) An analytic set in an arbitrary topological space 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 is a Hausdorff space, the last-mentioned condition is automatically satisfied. If is metrizable, this definition is equivalent to the classical one.
2) In a complete separable metric space the classical analytic sets are identical with -sets (cf. -set). This fact forms the base of another definition of analytic sets (in their capacity as -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 .
4) An analytic set is a continuous image of a set belonging to the family , where 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 ; -analytic sets are obtained from closed sets of a topological space by generalized -operations (cf. -operation; the Baire space of countable weight is replaced by a Baire space of weight ) and are a generalization of analytic sets in the sense of 2).