[FOM] Polish spaces and descriptive set theory

joeshipman@aol.com joeshipman at aol.com
Fri Sep 7 16:23:45 EDT 2007

----Original Message-----
From: Jan Pax <pax0 at seznam.cz>

Can someone please give me the reason why in descriptive set theory we
prefer to work with perfect Polish spaces
(i.e. toplogical spaces homeomorphic to a complete separable metric 
without isolated points)
like Baire space w^w, instead of the more intuitive real line?

The real line is also a complete separable metric space without 
isolated points, but it has properties which, while useful for 
analysis, are irrelevant for measure theory and logic. Baire space has 
the property that it is homeomorphic to its own square, which greatly 
simplifies the study of the kinds of phenomena descriptive set 
theorists care about (while collapsing and trivializing phemomena like 
"dimension" which they don't care so much about).

-- JS
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