Polish space | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-03-27

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Polish_space


00:01:52 1 Properties
00:03:55 2 Characterization
00:05:27 3 Polish metric spaces
00:06:31 4 Generalizations of Polish spaces
00:06:43 4.1 Lusin spaces
00:07:49 4.2 Suslin spaces
00:09:02 4.3 Radon spaces
00:09:51 4.4 Polish groups
00:11:25 5 See also



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SUMMARY
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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 topologists and logicians—Sierpiński, Kuratowski, Tarski and others.  However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
Common examples of Polish spaces are the real line, any separable Banach space, the Cantor space, and Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish; e.g., the open interval (0, 1) is Polish.
Between any two uncountable Polish spaces, there is a Borel isomorphism; that is, a bijection that preserves the Borel structure.  In particular, every uncountable Polish space has the cardinality of the continuum.
Lusin spaces, Suslin spaces, and Radon spaces are generalizations of Polish spaces.