
Re: G_delta
Posted:
Jan 14, 2013 5:10 AM


On Jan 14, 2:52 am, William Elliot <ma...@panix.com> wrote: > On Mon, 14 Jan 2013, Butch Malahide 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. > > Yes, that's correct. In fact it's true for regular Lindeloff spaces > for which every point is a G_delta (whence the G_delta question). > > If S is comapct Hausdorff, then every point is G_delta iff S is 1st > countable. If S is countable, then every point is G_delta. > Thus the problem is to find h for uncountable, not 1st countable S. > > > 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? > > Yes, > e:S > [0,1]^F, x > prod{ f(x)  f in F } > where > F = C(S,[0,1]) > is an embedding for any Tychonov T0 space S. > > How can that be used?
Use the fact that a mapping into a product space is continuous if and only if each of its projections is continuous. The projections are realvalued functions, and so each of them can be extended continuously from omega_1 to omega_1 + 1.

