In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element of some perfect Polish space. In practice, a pointclass is usually characterized by some sort of definability property; for example, the collection of all open sets in some fixed collection of Polish spaces is a pointclass. (An open set may be seen as in some sense definable because it cannot be a purely arbitrary collection of points; for any point in the set, all points sufficiently close to that point must also be in the set.)
Pointclasses find application in formulating many important principles and theorems from set theory and real analysis. Strong set-theoretic principles may be stated in terms of the determinacy of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as Lebesgue measurability (and indeed universal measurability), the property of Baire, and the perfect set property.
In the mathematical field of descriptive set theory, a pointclass is a collection of sets of points, where a point is ordinarily understood to be an element...
In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursive pointsets and is closed under recursive...
pointclass. Equivalently, for each ordinal α ≤ θ the collection Wα of sets that show up before stage α is a pointclass. Conversely, every pointclass is...
(perfect-information game) determinacy for a boldface pointclass implies Blackwell determinacy for the pointclass. This, combined with the Borel determinacy theorem...
If Γ {\displaystyle {\boldsymbol {\Gamma }}} is an adequate pointclass whose dual pointclass has the prewellordering property, then Γ {\displaystyle {\boldsymbol...
The converse does not hold; however, if every game in a given adequate pointclass Γ {\displaystyle \Gamma } is determined, then every set in Γ {\displaystyle...
n, together with a real parameter. The inductive sets form a boldface pointclass; that is, they are closed under continuous preimages. In the Wadge hierarchy...
difference hierarchy over a pointclass is a hierarchy of larger pointclasses generated by taking differences of sets. If Γ is a pointclass, then the set of differences...
Point class may refer to Pointclass sets in mathematics Point-class sealift ship Point-class cutter This disambiguation page lists articles associated...
Kripke–Platek set theory and second-order arithmetic. This box: view talk edit Pointclass Prewellordering Scale property Kechris, Alexander S. (1994). Classical...
lemma may be expressed generally as follows: Let Γ be a non-selfdual pointclass closed under real quantification and ∧, and ≺ a Γ-well-founded relation...
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