Re: Closed set with Cantor-Bendixson Rank omega

poopdeville_at_gmail.com
Date: 02/10/05 

Date: 9 Feb 2005 16:18:55 -0800

Mike Oliver wrote:
> poopdeville@gmail.com wrote:
>
> > Does anyone know of an example of a closed set in a Polish space
with
> > Cantor-Bendixson rank \omega?
>
> Just working it out on the back of my eyeballs, I think you
> could just take a set of rationals (considered as a subset
> of the reals) having order-type omega^omega. Haven't checked
> it carefully.

I'm not sure I follow. What does "order-type omega^omega" mean? Is
this lexicographic order on sequences? My first impulse was to
consider sets of the form {1/a + 1/b: a, b in N}, but that only gets me
a C-B rank = 2. 1/a_1 + 1/a_2 + 1/a_3 gives three, and intuitively, it
seems that we can form these doodles for any finite rank. I guessed
that maybe the range of f:N^N -> R defined by f({x_n}) = Sum (1/x_n)^2
would work, but proving it looks rough.

'cid 'ooh