Re: When are two reals not equal?
herbzet
herbzet wrote:
> reas...@gmail.com wrote:
> > Define a closed interval of rational numbers:
> >
> > A_0 = [x_0, y_0]
> >
> > where x_0 = -1/2 and y_0 = 1/2.
>
> This set is not closed, since it does not contain all of
> its limit points.
Using '^' for intersection, Wikepedia says at
http://en.wikipedia.org/wiki/Closed_set#Examples_of_closed_sets
"... the set [0,1] ^ Q of rational numbers between 0 and 1 (inclusive)
is closed in the space of rational numbers"
which I don't understand, since there are Cauchy convergent sequences
in [0, 1] ^ Q that have no limit in Q.
--
hz
Mariano Suárez-Alvarez
> herbzet wrote:
It says that that set is closed _in_the_space_of_rational_numbers_.
By definition, a subset S of Q is closed _in_Q_ if there exists
a closed subset T of R such that S = T ^ Q.
-- m
Tim Little
> http://en.wikipedia.org/wiki/Closed_set#Examples_of_closed_sets
>
> "... the set [0,1] ^ Q of rational numbers between 0 and 1 (inclusive)
> is closed in the space of rational numbers"
Yes: for every sequence in [0,1]^Q that has a limit *in Q*, that limit
is also within [0,1]^Q. [0,1]^Q is not closed in R, because there are
sequences in [0,1]^Q with limits in R that are not in [0,1]^Q.
> which I don't understand, since there are Cauchy convergent sequences
> in [0, 1] ^ Q that have no limit in Q.
Yes, Q is not complete.
- Tim
µ
You make a confusion between "closed" and "complete".
The set Q ^ [0,1] is closed in Q although not complete.
There is no contradiction, since there is a theorem "if A is complete
and B closed in A then B is complete", you just have found a
counter-example for the converse.
David C. Ullrich
>
>
>herbzet wrote:
>> reas...@gmail.com wrote:
>
>> > Define a closed interval of rational numbers:
>> >
>> > A_0 = [x_0, y_0]
>> >
>> > where x_0 = -1/2 and y_0 = 1/2.
>>
>> This set is not closed, since it does not contain all of
>> its limit points.
>
>Using '^' for intersection, Wikepedia says at
>
>http://en.wikipedia.org/wiki/Closed_set#Examples_of_closed_sets
>
>"... the set [0,1] ^ Q of rational numbers between 0 and 1 (inclusive)
>is closed in the space of rational numbers"
That's true. Say E = [0,1] ^ Q. Saying that E is closed _in Q_ means
this: If (x_n) is a sequence in E, x_n converges to x, _and_ x is in Q
then x is in E.
>which I don't understand, since there are Cauchy convergent sequences
>in [0, 1] ^ Q that have no limit in Q.
So what? That just says that E is not _complete_. Complete and closed
are not the same thing. (They're equivalent for subsets of a complete
space, but not for subsets of _any_ space.)
David C. Ullrich
Dave Seaman
> herbzet wrote:
>> reas...@gmail.com wrote:
>> > Define a closed interval of rational numbers:
>> >
>> > A_0 = [x_0, y_0]
>> >
>> > where x_0 = -1/2 and y_0 = 1/2.
>>
>> This set is not closed, since it does not contain all of
>> its limit points.
It contains all of its limit points in Q, which is a subspace of R.
If X is a topological space and Y is a subspace, the closed sets in Y (in the relative
topology) are exactly the intersections with Y of the closed sets in X.
> Using '^' for intersection, Wikepedia says at
> http://en.wikipedia.org/wiki/Closed_set#Examples_of_closed_sets
> "... the set [0,1] ^ Q of rational numbers between 0 and 1 (inclusive)
> is closed in the space of rational numbers"
> which I don't understand, since there are Cauchy convergent sequences
> in [0, 1] ^ Q that have no limit in Q.
That means Q is not complete. It has nothing to do with whether Q is closed.
--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
herbzet
Dave Seaman wrote:
>
> On Sat, 09 Feb 2008 22:31:53 -0500, herbzet wrote:
>
> > herbzet wrote:
> >> reas...@gmail.com wrote:
>
> >> > Define a closed interval of rational numbers:
> >> >
> >> > A_0 = [x_0, y_0]
> >> >
> >> > where x_0 = -1/2 and y_0 = 1/2.
> >>
> >> This set is not closed, since it does not contain all of
> >> its limit points.
>
> It contains all of its limit points in Q, which is a subspace of R.
>
> If X is a topological space and Y is a subspace, the closed sets in Y (in the relative
> topology) are exactly the intersections with Y of the closed sets in X.
>
> > Using '^' for intersection, Wikepedia says at
>
> > http://en.wikipedia.org/wiki/Closed_set#Examples_of_closed_sets
>
> > "... the set [0,1] ^ Q of rational numbers between 0 and 1 (inclusive)
> > is closed in the space of rational numbers"
>
> > which I don't understand, since there are Cauchy convergent sequences
> > in [0, 1] ^ Q that have no limit in Q.
>
> That means Q is not complete. It has nothing to do with whether Q is closed.
Thanks to everyone for pointing out the closed/complete distinction.
--
hz