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

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David C. Ullrich

Posts: 21,553
Registered: 12/6/04
Re: G_delta
Posted: Jan 14, 2013 9:49 AM
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On Mon, 14 Jan 2013 00:23:08 -0800 (PST), Butch Malahide
<> wrote:

>On Jan 14, 12:04 am, William Elliot <> 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...

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