
Re: G_delta
Posted:
Jan 21, 2013 3:59 AM


On Sun, 20 Jan 2013, Butch Malahide wrote: > On Jan 20, 12:14 am, William Elliot <ma...@panix.com> wrote: > > > > Your proof generaly follows the proof for f in C(omega_1,S), > > > where S is regular Lindelof and ever point is G_delta, that > > f is eventually constant. There are some differences in the > > premises of the two theorems that I'm going to puzzle upon > > and try to harmonize. > > The puzzle is whether I needed to put that silly adverb "hereditarily" > in front of "Lindelof". Now that you mention it, just plain Lindelof > is good enough. I only used "hereditarily Lindelof" to show that Z is > Lindelof. It's easy to see (as shown in the last step of the argument > I posted) that Y\Z contains at most one point. It follows from the > other assumptions that Y\Z is a G_{delta}, i.e., Z is an F_{sigma} in > Y; and of course an F_{sigma} subspace of a Lindelof space is Lindelof.
Ok, the other difference is a) regular & every point a G_delta and b) every point the countable intersection of closed nhoods of the point.
That is for all p, there's some closed Kj, j in N with for all j in N, p in int Kj & {p} = /\_j Kj.
Yes, a) > b) but I doubt the converse. In other words, is a) strictly stonger than b).
Now if we assume a Lindelof space, is a) still strictly stronger than b?
I think it is. Are you of the same opinion?

