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Topic: G_delta
Replies: 28   Last Post: Jan 26, 2013 3:50 AM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: G_delta
Posted: Jan 19, 2013 2:51 AM
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On Jan 19, 1:05 am, William Elliot <ma...@panix.com> wrote:
> On Tue, 15 Jan 2013, Butch Malahide wrote:
> > On Jan 14, 11:10 pm, William Elliot <ma...@panix.com> wrote:
>
> > > Does this generalize to every uncountable limit ordinal eta,
> > > that f in C(eta,R) is eventually constant and thusly the Cech
> > > Stone compactification of of eta is eta + 1?  Does eta need
> > > to have an uncountable cofinality for this generalization?

>
> > Yes, the same argument that works for omega_1 also works for any
> > ordinal of uncountable cofinality.

>
> > Let X be a linearly ordered topological space (i.e., a linearly
> > ordered set with its order topology) in which every increasing
> > sequence converges. [Examples: any ordinal of uncountable cofinality;
> > the long line; any countably compact LOTS.] Call a subset of X
> > "bounded" if it has an *upper* bound in X, "unbounded" otherwise.
> > Observe that (1) the union of countably many bounded sets is bounded,
> > and (2) the intersection of countably many unbounded closed sets is
> > unbounded.

>
> Does not (1) hold because every increasing sequence converges?
> Wouldn't every countable set has an upper bound suffice?


Yes, "every countable set has an upper bound" would suffice for (1),
but not for (2).

> > Let Y be a topological space which is hereditarily Lindelof and such
> > that, for each point y in Y, the set {y} is the intersection of
> > countably many closed neighborhoods of y. [Example: any separable
> > metric space.]

>
> > THEOREM. If X and Y are as stated above, then every function f in
> > C(X,Y) is eventually constant.

>
> > PROOF. We may assume that X has no greatest element. For S a subset of
> > Y, let g(S) = {x in X: f(x) is in S}. Let Z = {y in Y: g({y}) is
> > bounded}.

>
> Is assuming X has no greatest element, an additional premise?


No. The assumption that X has no greatest element is made without loss
of generality, because the contrary case is trivial: if X has a
greatest element, then every function with domain X is eventually
constant.

> Is Z closed?

It will be shown that Y\Z contains a single point c, the eventual
value of the function f, and so Z = Y\{c}. To ask whether Z is closed
is to ask whether c is an isolated point of Y. It could, but it
doesn't have to be.

> > CLAIM. Each point z in Z has a neighborhood V_z such that g(V_z) is
> > bounded.

>
> > PROOF OF CLAIM. Let {U_n: n in N} be a countable collection of closed
> > neighborhoods of z whose intersection is {z}. Assume for a
> > contradiction that each set g(U_n) is unbounded. Since f is
> > continuous, each g(U_n) is an unbounded closed set. By property (2)
> > above, g({z}) = /\{g(U_n): n in N} is unbounded, contradicting the
> > assumption that z is in Z.

>
> > Thus the set Z is covered by open sets V_z such that g(V_z) is
> > bounded. Since Y is hereditarily Lindelof, it follows that Z is
> > covered by countably many open sets V such that g(V) is bounded. In
> > view of property (1)  above, it follows that g(Z) is bounded. Since
> > g(Y) = X is unbounded, Y\Z is nonempty. All we have left to show is
> > that Y\Z consists of a single point. Assume for a contradiction that Y
> > \Z contains two distinct points c and d. Thus g({c}) and g({d}) are
> > unbounded closed subsets of X. By property (2) above, the intersection
> > of g({c}) and g({d}) is unbounded; in particular, it is nonempty. Let
> > x be a point in the intersection of g({c}) and g({d}). Then c = f(x) =
> > d, contradicting the assumption that c and d are two distinct points.




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