Algebraic numbers are not G_delta ?
jane
Thanks.
quasi
>Is it true that the set of algebraic numbers in R is not G_delta
>(countable intersection of open sets) ?
Yes.
Let A be any countable, dense subset of R, and let B=R\A.
Suppose A is G-delta.
Then B is F-sigma, a countable union of closed sets, all of which must
be nowhere dense (since A is dense).
But then R would be a countable union of nowhere dense sets, (take the
ones whose union is B together with all singletons {a} where a is in
A), contrary to the Baire Category Theorem.
quasi
Dave L. Renfro
.
>> Is it true that the set of algebraic numbers in R is
>> not G_delta (countable intersection of open sets) ?
quasi wrote:
.
> Yes.
>
> Let A be any countable, dense subset of R, and let B=R\A.
>
> Suppose A is G-delta.
[snip rest]
Indeed, any countable G_delta set is "maximally small"
in a certain topological sense. More specifically,
a countable G_delta set in R (or in any complete
separable metric space) is nowhere dense relative
to every nonempty perfect subset of R. This means,
for instance, that not only the set itself but
also the closure of the set fails to be dense
in any portion of any perfect set, regardless
of how small/thin the perfect set is. Moreover,
this characterizes the countable G_delta sets
(often called "scattered sets" in the literature),
since any set that is nowhere dense relative to
every nonempty perfect set is a countable G_delta
set. An example of a situation in which scattered
sets arise is the set of all points where a function
has an infinite limit. Given f:R --> R, the set
{y: lim(x --> y) f(x) is +infinity} is scattered.
In fact, any scattered subset of R can be realized
in this way for some function f. The same is true
if the limit operation is restricted to being a
left limit or to being a right limit, and/or
+infinity being replaced with -infinity or with
either infinity. However, if "infinity" is replaced
with a specified finite value, the set may no
longer be countable (e.g. a constant function),
although it will still be G_delta.
Being scattered is a type of smallness that fits
between isolated sets (all of which are scattered,
but a convergent sequence union its limit point is
a scattered set that isn't an isolated set) and
the nowhere dense sets (every scattered set is
nowhere dense, but a nonempty nowhere dense perfect
set -- the Cantor set, for instance -- is a nowhere
dense set that isn't scattered.
Dave L. Renfro