Perfect Polish space

poopdeville_at_gmail.com
Date: 03/04/05 

Date: 4 Mar 2005 15:56:38 -0800

My questions seem to be getting more and more obscure. Here goes:

Starting for a particular subset of my perfect set, I've managed to
come up with finite increasing sequences of perfect and nowhere
perfects sets. No matter what the answer is, it seems to me that it
will rely on the Cantor-Bendixon theorem, but it isn't straightforward
at all, as perfect sets very often contain open subsets.

Can a perfect set in a Polish space be the disjoint union of a perfect
set and a countable open set? (Are there even countable open sets in
Polish topologies? Thinking about countable open sets in completely
metrizable topologies makes my head hurt too much to figure it out).

As always, references and (counter-)examples are appreciated.

Thanks,
'cid 'ooh