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