This is often because the ground field has additional structure that isn't captured by analytic sets.

This article is about analytic sets as defined in descriptive set theory.

Care is required to distinguish this usage from the term analytic set which has a different meaning.

Together they began the study of analytic sets.

The complement of an analytic set need not be analytic.

An analytic variety is also called a (real or complex) analytic set.

See chapter 1, paragraph 2 Definition and simplest properties of analytic sets.

There are several equivalent definitions of analytic set.

Every analytic set has the perfect set property.

In fact, it is analytic, and complete in the class of analytic sets.

If contains an analytic set that is not Borel-separated from , then .

Analytic sets are also called (see projective hierarchy).

The proof that the début is measurable is rather involved and involves properties of analytic sets.

Every analytic set is universally measurable.

See analytic set.

Together with his student Mikhail Yakovlevich Suslin, he developed the theory of analytic sets.

Similarly for subsets: if one representative of a germ is an analytic set then so are all representatives, at least on some neighbourhood of x.

Just beyond the Borel sets in complexity are the analytic sets and coanalytic sets.

He contributed greatly to the theory of analytic sets, sometimes called after him, a kind of a set of reals which is definable via trees.

Furthermore, the theory of R-functions allow conversions of such representations into a single function inequality for any closed semi analytic set.

Suslin proved that if the complement of an analytic set is analytic then the set is Borel.

Therefore the image of an analytic set under an analytic map is not necessarily an analytic set.

This can be avoided by working with subanalytic sets, which are much less rigid than analytic sets but which are not defined over arbitrary fields.

An alternative characterization, in the specific, important, case that is Baire space, is that the analytic sets are precisely the projections of trees on .

In complex analysis, a field in mathematics, the Remmert-Stein theorem, introduced by gives conditions for the closure of an analytic set to be analytic.