Descriptive set theory: Difference between revisions

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* If ''A''_''n'' is a Borel set for each natural number ''n'', then the union <math>\bigcup A_n</math> is a Borel set. That is, the Borel sets are closed under countable unions.

A fundamental result shows that any two uncountable Polish spaces ''X'' and ''Y'' are '''[[Borel isomorphism|Borel isomorphic''']]: there is a bijection from ''X'' to ''Y'' such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This gives additional justification to the practice of restricting attention to Baire space and Cantor space, since these and any other Polish spaces are all isomorphic at the level of Borel sets.

=== Borel hierarchy ===