
Re: G_delta
Posted:
Jan 14, 2013 2:16 PM


On Jan 14, 8:49 am, David C. Ullrich <ullr...@math.okstate.edu> wrote: > On Mon, 14 Jan 2013 00:23:08 0800 (PST), Butch Malahide > > > > > > <fred.gal...@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... Hide quoted text 
Thanks!

