Revision as of 19:53, 3 April 2008 edit Mrw7 (talk \| contribs) 46 edits same cardinality required for Polish spaces to be Borel isomorphic ← Previous edit		Revision as of 19:56, 3 April 2008 edit undo Trovatore (talk \| contribs) Autopatrolled, Extended confirmed users, Pending changes reviewers 39,140 edits the result is much stronger than that (and countable Polish spaces aren't very interesting in the Borel category, so the previous version was almost right) Next edit →
Line 23: * If ''A''<sub>''n''</sub> 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'' ~~of the same cardinality~~ are '''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.

Descriptive set theory: Difference between revisions