Pointclass

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.

Read more about Pointclass:  Basic Framework, Boldface Pointclasses, Lightface Pointclasses

Other articles related to "pointclass":

Structure of The Wadge Hierarchy
... As for the Wadge lemma, this holds for any pointclass Γ, assuming the axiom of determinacy ... the collection of all sets strictly below on the Wadge hierarchy, this forms a pointclass ... ordinal α≤θ the collection Wα of sets which show up before stage α is a pointclass ...
Determinator - More General Games - Games of Imperfect Information
... game) determinacy for a boldface pointclass implies Blackwell determinacy for the pointclass ... that Blackwell determinacy for a boldface pointclass in turn implies ordinary determinacy for that pointclass), but as of 2010, it has not been proven that Blackwell determinacy implies perfect-information ...
Determinacy - More General Games - Games of Imperfect Information
... game) determinacy for a boldface pointclass implies Blackwell determinacy for the pointclass ... that Blackwell determinacy for a boldface pointclass in turn implies ordinary determinacy for that pointclass), but as of 2010, it has not been proven that Blackwell determinacy implies perfect-information-game ...
Lightface Pointclasses
... A set A is lightface if it is a union of a computable sequence of sets (that is, there is a computable enumeration of indices of sets such that A is the union of these sets) ... This relationship between lightface sets and their indices is used to extend the lightface Borel hierarchy into the transfinite, via recursive ordinals ...