In descriptive set theory

Descriptive set theory

In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...

, a subset of a Polish space

Polish space

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...

  is an analytic set if it is a continuous image of a Polish space. These sets were first defined by and his student .

Definition

There are several equivalent definitions of analytic set. The following conditions on a subspace A of a Polish space are equivalent:

• A is analytic.
• A is empty or a continuous image of the Baire space
  Baire space (set theory)
  In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called “reals.” It is often denoted B, N'N, or ωω...
  
   ω^ω.
• A is a Suslin space, in other words A is the image of a Polish space under a continuous mapping.
• A is the continuous image of a Borel set in a Polish space.
• A is a Suslin set
  Suslin set
  The concept of a Suslin set was first used by Mikhail Yakovlevich Suslin when he was researching the properties of projections of Borel sets in \R^2 onto the real axis...
  
  , the image of the Suslin operation.
• There is a Polish space and a Borel
  Borel algebra
  In mathematics, a Borel set is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement...
  
   set such that is the projection of ; that is,

• A is the projection of a closed set in X times the Baire space.
• A is the projection of a G_δ set in X times the Cantor space.

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

Properties

Analytic subsets of Polish spaces are closed under countable unions and intersections, continuous images, and inverse images.
The complement of an analytic set need not be analytic. Suslin proved that if the complement of an analytic set is analytic then the set is Borel. (Conversely any Borel set is analytic and Borel sets are closed under complements.) Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there is a Borel set containing one and disjoint from the other. This is sometimes called the "Luzin separability principle" (though it was implicit in the proof of Suslin's theorem).

Analytic sets are always Lebesgue measurable (indeed, universally measurable) and have the property of Baire and the perfect set property

Perfect set property

In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset...

.

Projective hierarchy

Analytic sets are also called (see projective hierarchy). Note that the bold font in this symbol is not the Wikipedia convention, but rather is used distinctively from its lightface counterpart (see analytical hierarchy

Analytical hierarchy

In mathematical logic and descriptive set theory, the analytical hierarchy is a higher type analogue of the arithmetical hierarchy. It thus continues the classification of sets by the formulas that define them.-The analytical hierarchy of formulas:...

). The complements of analytic sets are called coanalytic sets, and the set of coanalytic sets is denoted by .
The intersection is the set of Borel sets.