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Topic: G_delta
Replies: 28   Last Post: Jan 26, 2013 3:50 AM

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William Elliot

Posts: 1,449
Registered: 1/8/12
Re: G_delta
Posted: Jan 16, 2013 4:57 AM
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On Tue, 15 Jan 2013, Butch Malahide wrote:
> On Jan 15, 3:22 am, William Elliot <ma...@panix.com> wrote:

> > > > Does this generalize to every uncountable limit ordinal eta,
> > > > that f in C(eta,R) is eventually constant and thusly the Cech
> > > > Stone compactification of of eta is eta + 1? Does eta need
> > > > to have an uncountable cofinality for this generalization?

> >
> > > I'd expect cofinality to have a lot to do with it, wouldn't you?
> >
> > Indeed, f in C(omega_1 + omega_0, R) is not eventually constant.
> >
> > If eta has finite cofinality, then any f in C(eta,R) is eventaully
> > constant, namely from the last element of eta onward.  That however
> > isn't the full story for if f in C(omega_1 + 1, R), then f is
> > constant from some xi < omega_1 and not just from the last element.
> >
> > If eta has denumberable cofinality, does this work to show
> > f in C(eta,R) is eventually constant?

>
> What does "denumberable" mean? Is that a real word, or did you make it


It's a mispelling of denumerable.

> up? Does omega_1 + omega_0 have "denumberable cofinality"?
>

It has denumerable cofinality.

> > Let (aj)_j be an increaing sequence within eta with denumberable
> > cofinality.

>
> Just any old increasing sequence within eta? You don't care if it
> converges to eta or not? In that case why not simply set aj = j?


No it has to be cofinal with eta, ie converging to eta.

> > Let K = { aj + 1 | j in N }.
> > Then f(eta\K) = {0}, f(aj) = j, j in N is in C(eta,R)
> > and isn't eventually constant.


> Is this supposed to be the answer to the question you asked in the
> previous paragraph?


Yes.

> Why did you ask the question if you knew the answer?

I don't. I was attempting a proof of which I wan't sure.

> However, your function f is not well-defined. What is f(a1)? On the
> one hand, you say that f(aj) = j for j in N, so f(a1) = 1. On the
> other hand, f(a1) = 0 since a1 is an element of the set eta\K.


Whoops. f(eta\K) = {0}, f(aj + 1) = j, j in N is in C(eta,R).

Ok, is that as corrected a proof showing ordinals with denumerable
cofinality aren't pseudocompact? For example, omega_(omega_0).

> > What happens went the cofinality of eta is uncountable?
> > Is f in C(eta,R) eventually constant?

>
> Seems plausible enough. Are you saying that the method of proof used
> for omega_1 doesn't work for ordinals of uncountable cardinality?
> Where does it break down?


In the use of cardinality properties. As both omega_1 and
omega_1 + omega_0 have the same cardinaly but different
results, it's seems unlikely without drafting some cofanilty
into the proof, it would be useable.



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