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

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William Elliot

Posts: 1,243
Registered: 1/8/12
Re: G_delta
Posted: Jan 14, 2013 1:04 AM
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On Sun, 13 Jan 2013, Butch Malahide wrote:
> On Jan 13, 11:09 pm, William Elliot <ma...@panix.com> wrote:

> > If p is a point in a compact Hausdorff space, is {p} a G_delta set?
> Not necessarily.
>
> Example: [0, omega_1] with the order topology. {omega_1} is not a
> G_delta set.


Oh of course, and that was the very space I was wrestling with.

> Example: The Tychonoff product of uncountably many 2-point discrete
> spaces. No singleton is a G_delta set.


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.

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?



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