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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 2:16 PM
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On Jan 14, 8:49 am, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> On Mon, 14 Jan 2013 00:23:08 -0800 (PST), Butch Malahide
>
>
>
>
>
> <fred.gal...@gmail.com> 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.

>
> >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?

>
> Seems right to me. In fact I believe it's right even with the
> terminal "or something like that" omitted...- Hide quoted text -


Thanks!



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