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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 3:23 AM
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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?



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