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

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William Elliot

Posts: 1,533
Registered: 1/8/12
Re: G_delta
Posted: Jan 15, 2013 4:22 AM
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On Mon, 14 Jan 2013, Butch Malahide wrote:
> On Jan 14, 11:10 pm, 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?

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

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



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