
Re: G_delta
Posted:
Jan 14, 2013 9:49 AM


On Mon, 14 Jan 2013 00:23:08 0800 (PST), Butch Malahide <fred.galvin@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 195758. 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 realvalued 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...

