Re: collections of well-ordered closed (or else F_sigma) subsets of the unit interval [0, 1]

David Bernier wrote:

This is interesting. It's hard to visualize what
a typical countable G_delta set for [0, 1]
looks like ...

Countable G_delta sets are the same as the scattered
sets (the countable "left-over part" you get from the
Cantor-Bendixson theorem), so they're identical to the
sets which have a countable Cantor-Bendoxson rank.

Some other characterizations of scattered sets
(with plenty of references) can be found in my posts
in the following thread from a few months ago:

sci.math -- Scattered sets are G-delta (May 2008)
http://groups.google.com/group/sci.math/browse_thread/thread/b1921d85ed192cb1

Note that some of my later posts give corrections to
earlier posts, so make sure you look through all of
them. Here are some comments I made in another thread
(sci.math, 14 February 2007): "Sets with countable
closure are rather "small" as countable sets go.
The so-called scattered subsets of the reals can
be somewhat larger, being characterized as the
countable G_delta sets, but even the scattered
sets are quite small. Scattered sets not only fail
to be dense in every interval (i.e. they are nowhere
dense in the reals), scattered sets are even nowhere
dense relative to every nonempty perfect set."

NOTE: If a set has countable closure, then the
set is scattered. Proof: Let Z be a set such that
the closure of Z is countable. Then clearly Z is
also countable. Thus, it remains to show that Z
is G_delta. To see this, note that the closure
of Z, cl(Z), is a closed set, and hence is a G_delta
set. (Closed sets are G_delta sets in any metric
space.) Now note that the removal of a single point
from a set is the same as intersecting the set
with a certain open set (namely, the open set
that is the underlying space minus the single point),
and there are at most countably many points of cl(Z)
to remove before we're left with Z. (The intersection
of a G_delta set with countably many open sets is
easily seen to be an intersection of a certain
countable collection of open sets, and hence a
G_delta set.)

Dave L. Renfro
.

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