
Re: G_delta
Posted:
Jan 16, 2013 4:57 AM


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 welldefined. 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.

