Lightface hierarchy | Wikipedia audio article
Channel: wikipedia tts
Date: 05/06/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Pointclass


00:01:12 1 Basic framework
00:03:09 2 Boldface pointclasses
00:06:13 3 Lightface pointclasses
00:10:31 4 Summary



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Speaking Rate: 0.8944744183712184
Voice name: en-US-Wavenet-B


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- Socrates


SUMMARY
=======
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.