```Date: Jan 16, 2013 4:57 AM
Author: William Elliot
Subject: Re: G_delta

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 itIt'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 andomega_1 + omega_0 have the same cardinaly but differentresults, it's seems unlikely without drafting some cofaniltyinto the proof, it would be useable.
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