
Re: G_delta
Posted:
Jan 15, 2013 4:22 AM


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?

