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?