Pointclass

In the mathematical field of <a href="/facts/Descriptive_set_theory/ek0tvUXb">descriptive set theory</a>, a pointclass is a collection of <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> of <a href="/facts/Point_(mathematics)/6YPAvX2a">points</a>, where a point is ordinarily understood to be an element of some <a href="/facts/Perfect_set/wKUjfCkM">perfect</a> <a href="/facts/Polish_space/CkUFcGap">Polish space</a>. In practice, a pointclass is usually characterized by some sort of definability property; for example, the collection of all <a href="/facts/Open_set/QfTCIqMu">open sets</a> 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 <a href="/facts/Set_theory/1ptfsvUP">set theory</a> and <a href="/facts/Real_analysis/12WjlXCW">real analysis</a>. Strong set-theoretic principles may be stated in terms of the <a href="/facts/Determinacy/5GMp9B6z">determinacy</a> of various pointclasses, which in turn implies that sets in those pointclasses (or sometimes larger ones) have regularity properties such as <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measurability</a> (and indeed <a href="/facts/Universally_measurable_set/8Utsfxle">universal measurability</a>), the <a href="/facts/Property_of_Baire/G2Sk5Azz">property of Baire</a>, and the <a href="/facts/Perfect_set_property/S8A7wxcN">perfect set property</a>.

Pointclass open-in-new

Pointclass