Re: Union of Uncountable Totally Imperfect Sets

poopdeville_at_gmail.com
Date: 02/25/05 

Date: Fri, 25 Feb 2005 16:30:04 +0000 (UTC)

Ali Enayat wrote:

> Note, however, that:
>
> 1. By a classical theorem of Suslin (around 1917), such a set cannot
> be Borel or even a continuous image of a Borel set, i.e., every
> uncountable Borel set (or continuous image of a Borel set) contains a
> perfect subset.

This is very closely related to the work I'm doing now. At the moment
I'm trying to prove the Lusin-Purves theorem by considering the
transfinite (Hausdorff) residue of the projection of a Borel set in the
cartesian product of standard Borel spaces. It turns out that the
closure of the projection can be expressed as the disjoint union of a
perfect set and two totally imperfect uncountable sets, hence my hope
of a general result for these sorts of cases.

Of course, I can apply Souslin's (other) theorem I can quickly derive
the result, but I'm trying to find new technologies. I actually ended
up proving that a nowhere perfect uncountable set can't be Borel, but I
hadn't thought of the second result you mentioned. Thanks. :-)

'cid 'ooh