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Re: G_delta
Posted:
Jan 15, 2013 12:10 AM
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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 real-valued 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
> real-valued 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?
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