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Topic: G_delta
Replies: 28   Last Post: Jan 26, 2013 3:50 AM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: G_delta
Posted: Jan 14, 2013 5:10 AM
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On Jan 14, 2:52 am, William Elliot <ma...@panix.com> wrote:
> On Mon, 14 Jan 2013, Butch Malahide wrote:
> > On Jan 14, 12:04 am, William Elliot <ma...@panix.com> wrote:
>
> > > In 1st countable spaces, every point p is G_delta.
> > > In fact, if every point of a compact Hausdorff space S,
> > > is G_delta, then S is 1st countable.

>
> > Oh, right, I'd forgotten that.
>
> > > 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. I'm not a
> > topologist, but I took a course in topology back in 1957-58. I'm not
> > sure I remember enough topology to answer a technical question like
> > that, but I'll take a stab at it.
> > 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.



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