
Re: G_delta
Posted:
Jan 15, 2013 12:10 AM


On Mon, 14 Jan 2013, Butch Malahide wrote: > On Jan 14, 2:52 am, William Elliot <ma...@panix.com> wrote:
> > > > Define g:omega_1 > omega_1 + 1, eta > eta. > > > > Assume S is compact Hausdorff and f in C(omega_1, S). > > > > > > Is there some h in C(omega_1 + 1, S) with f = hg? > > > > > Are you just asking whether a continuous function from omega_1 to a > > > compact Hausdorff space S can always be extended to a continuous > > > function on omega_1 + 1? That sounds like it should be true.
> > > First suppose S = [0, 1]. (Gotta walk before you can run.) I believe > > > that a continuous realvalued function on omega_1 must be eventually > > > constant. (I may even remember how to prove that, but it's late and I > > > don't want to do any hard thinking now.) In that case, just map the > > > point omega_1 to the same constant, and everything is fine. > > > > Yes, that's correct. In fact it's true for regular Lindeloff spaces > > for which every point is a G_delta (whence the G_delta question). > > > > If S is comapct Hausdorff, then every point is G_delta iff S is 1st > > countable. If S is countable, then every point is G_delta. > > Thus the problem is to find h for uncountable, not 1st countable S. > > > > > Now, can't the result for an arbitrary compact Hausdorff space S be > > > derived from the [0, 1] case? Say, by embedding S in a cube, or > > > something like that? > > > > Yes, > > e:S > [0,1]^F, x > prod{ f(x)  f in F } > > where > > F = C(S,[0,1]) > > is an embedding for any Tychonov T0 space S. > > > > How can that be used? > > Use the fact that a mapping into a product space is continuous if and > only if each of its projections is continuous. The projections are > realvalued functions, and so each of them can be extended > continuously from omega_1 to omega_1 + 1. > Well joggle my mind. That all was worked out in Theorem 19.12 of General Topology by Stephen Willard:
If (g,Y) is a Hausdorff compactification of X, and g(X) is C* embedded in Y, then (g,Y) is universal.
In this case, g(omega_1) = omega_1 and if f in C(omega_1,R), then f is eventually constant, hence extendible to omega_1 + 1 (ie, omega_1 is C* embedded in omega_1 + 1).
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?

