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Guillaume Filteau
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 Posted: Mon Jun 16, 2008 5:07 am Post subject: Nested sequence of intervals and the empty set |
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I have a simple question:
If you have a nested sequence of intervals in R, why can't one of these intervals be the empty set ?
Thanks,
Guillaume |
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Guillaume Filteau
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| Posted: Mon Jun 16, 2008 5:56 am Post subject: Re: Nested sequence of intervals and the empty set |
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Thank you for your answers.
The "Nested Intervals Theorem" states that the intersection of a nested sequence of intervals in R will never be empty.
If the definitions make the empty set nested within itself, and if the empty set is an interval, then the Nested Intervals Theorem is false.
Guillaume |
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William Elliot
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| Posted: Mon Jun 16, 2008 10:30 am Post subject: Re: Nested sequence of intervals and the empty set |
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On Mon, 16 Jun 2008, Guillaume Filteau wrote:
| Quote: |
If you have a nested sequence of intervals in R, why can't one of these
intervals be the empty set ?
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Which way is the nested sequence going?
A1 subset A2 subset A3 ...
or
.... A3 subset A2 subset A1 ?
In the first case, if Aj = nulset, then for all k <= j, A_k = nulset.
In the other case, if Aj = nulset, then for all k => j, A_k = nulset.
In the all for all j, Aj = nulset, then either way, you've a nested
sequence of intervals (or subsets to generalize) soley of empty sets.
So of course, either way, if they are all empty, then you've a nested
sequence of intervals. |
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Virgil
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| Posted: Mon Jun 16, 2008 10:32 am Post subject: Re: Nested sequence of intervals and the empty set |
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In article
<3638218.1213592912598.JavaMail.jakarta@nitrogen.mathforum.org>,
Guillaume Filteau <filteau@unc.edu> wrote:
| Quote: |
I have a simple question:
If you have a nested sequence of intervals in R, why can't one of these
intervals be the empty set ?
Thanks,
Guillaume
|
If you are to have a sequence of strictly nested intervals, with no two
identical, then if one of them is empty (or for closed intervals , is a
single point) the sequence must end there.
Thus for infinite nested sequences, no interval can be the empty set. |
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Brian M. Scott
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| Posted: Mon Jun 16, 2008 10:35 am Post subject: Re: Nested sequence of intervals and the empty set |
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On Mon, 16 Jun 2008 01:07:54 EDT, Guillaume Filteau
<filteau@unc.edu> wrote in
<news:3638218.1213592912598.JavaMail.jakarta@nitrogen.mathforum.org>
in alt.math.undergrad:
| Quote: |
I have a simple question:
If you have a nested sequence of intervals in R, why can't
one of these intervals be the empty set ?
|
Whether it can or not depends on what definition of
'interval' you're using. If by 'interval' you simply mean
any set J of real numbers with the property that if x and y
belong to J, and x <= z <= y, then z belongs to J, then one
of them *can* be empty. Of course all of the later
intervals in the sequence must then also be empty, or they
wouldn't be nested. For instance, your sequence of
intervals could be <(0, 4), (1, 4), (1, 3), (2, 2), (2, 2),
(2, 2), ...>.
Brian |
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William Elliot
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| Posted: Mon Jun 16, 2008 11:00 am Post subject: Re: Nested sequence of intervals and the empty set |
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On Mon, 16 Jun 2008, Brian M. Scott wrote:
| Quote: |
On Mon, 16 Jun 2008 01:56:20 EDT, Guillaume Filteau
news:17423020.1213595810828.JavaMail.jakarta@nitrogen.mathforum.org
in alt.math.undergrad:
Thank you for your answers.
The "Nested Intervals Theorem" states that the
intersection of a nested sequence of intervals in R will
never be empty.
No, it doesn't: it says that the intersection of a nested
sequence of non-empty closed intervals in R is not empty.
If C is a nested chain of nonnul bounded closed subsets |
of R, then /\C is not empty.
It's a property of compactness that you're studying.
C can be countable, uncountable. Even not a sequence.
{ [-1/n, 1/n], [-(1+1/n), 1+1/n] | n in N }
The elements of C don't have to be intervals. |
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Paul Sperry
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| Posted: Mon Jun 16, 2008 11:00 am Post subject: Re: Nested sequence of intervals and the empty set |
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In article
<17423020.1213595810828.JavaMail.jakarta@nitrogen.mathforum.org>,
Guillaume Filteau <filteau@unc.edu> wrote:
| Quote: |
Thank you for your answers.
The "Nested Intervals Theorem" states that the intersection of a nested
sequence of intervals in R will never be empty.
If the definitions make the empty set nested within itself, and if the empty
set is an interval, then the Nested Intervals Theorem is false.
Guillaume
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See <http://en.wikipedia.org/wiki/Nested_intervals>
--
Paul Sperry
Columbia, SC (USA) |
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Brian M. Scott
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| Posted: Mon Jun 16, 2008 11:00 am Post subject: Re: Nested sequence of intervals and the empty set |
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On Mon, 16 Jun 2008 01:56:20 EDT, Guillaume Filteau
<filteau@unc.edu> wrote in
<news:17423020.1213595810828.JavaMail.jakarta@nitrogen.mathforum.org>
in alt.math.undergrad:
| Quote: |
Thank you for your answers.
The "Nested Intervals Theorem" states that the
intersection of a nested sequence of intervals in R will
never be empty.
|
No, it doesn't: it says that the intersection of a nested
sequence of non-empty closed intervals in R is not empty.
[...]
Brian |
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The World Wide Wade
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| Posted: Thu Jun 19, 2008 11:00 am Post subject: Re: Nested sequence of intervals and the empty set |
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In article <1e2g7oyhx3ikm.sfc98i1qz8cz.dlg@40tude.net>,
"Brian M. Scott" <b.scott@csuohio.edu> wrote:
| Quote: |
On Mon, 16 Jun 2008 01:56:20 EDT, Guillaume Filteau
filteau@unc.edu> wrote in
news:17423020.1213595810828.JavaMail.jakarta@nitrogen.mathforum.org
in alt.math.undergrad:
Thank you for your answers.
The "Nested Intervals Theorem" states that the
intersection of a nested sequence of intervals in R will
never be empty.
No, it doesn't: it says that the intersection of a nested
sequence of non-empty closed intervals in R is not empty.
|
You'll need those to be bounded.
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Virgil
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| Posted: Thu Jun 19, 2008 11:00 am Post subject: Re: Nested sequence of intervals and the empty set |
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In article
<aderamey.addw-832CE7.23025818062008@newsgroups.comcast.net>,
The World Wide Wade <aderamey.addw@comcast.net> wrote:
| Quote: |
In article <1e2g7oyhx3ikm.sfc98i1qz8cz.dlg@40tude.net>,
"Brian M. Scott" <b.scott@csuohio.edu> wrote:
On Mon, 16 Jun 2008 01:56:20 EDT, Guillaume Filteau
filteau@unc.edu> wrote in
news:17423020.1213595810828.JavaMail.jakarta@nitrogen.mathforum.org
in alt.math.undergrad:
Thank you for your answers.
The "Nested Intervals Theorem" states that the
intersection of a nested sequence of intervals in R will
never be empty.
No, it doesn't: it says that the intersection of a nested
sequence of non-empty closed intervals in R is not empty.
You'll need those to be bounded.
|
WHY? |
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David C. Ullrich
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| Posted: Thu Jun 19, 2008 11:00 am Post subject: Re: Nested sequence of intervals and the empty set |
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On Thu, 19 Jun 2008 00:25:01 -0600, Virgil <Virgil@gmale.com> wrote:
| Quote: |
In article
aderamey.addw-832CE7.23025818062008@newsgroups.comcast.net>,
The World Wide Wade <aderamey.addw@comcast.net> wrote:
In article <1e2g7oyhx3ikm.sfc98i1qz8cz.dlg@40tude.net>,
"Brian M. Scott" <b.scott@csuohio.edu> wrote:
On Mon, 16 Jun 2008 01:56:20 EDT, Guillaume Filteau
filteau@unc.edu> wrote in
news:17423020.1213595810828.JavaMail.jakarta@nitrogen.mathforum.org
in alt.math.undergrad:
Thank you for your answers.
The "Nested Intervals Theorem" states that the
intersection of a nested sequence of intervals in R will
never be empty.
No, it doesn't: it says that the intersection of a nested
sequence of non-empty closed intervals in R is not empty.
You'll need those to be bounded.
WHY?
|
BECAUSE OTHERWISE THE STATEMENT IS FALSE.
Say I_n is the closed interval [n, infinity). What's the intersection
of the I_n?
David C. Ullrich |
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Virgil
Guest
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| Posted: Fri Jun 20, 2008 1:46 am Post subject: Re: Nested sequence of intervals and the empty set |
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In article <6u0k5491c7ko4n6dkicahigclc1tmfruv9@4ax.com>,
David C. Ullrich <dullrich@sprynet.com> wrote:
| Quote: |
On Thu, 19 Jun 2008 00:25:01 -0600, Virgil <Virgil@gmale.com> wrote:
In article
aderamey.addw-832CE7.23025818062008@newsgroups.comcast.net>,
The World Wide Wade <aderamey.addw@comcast.net> wrote:
In article <1e2g7oyhx3ikm.sfc98i1qz8cz.dlg@40tude.net>,
"Brian M. Scott" <b.scott@csuohio.edu> wrote:
On Mon, 16 Jun 2008 01:56:20 EDT, Guillaume Filteau
filteau@unc.edu> wrote in
news:17423020.1213595810828.JavaMail.jakarta@nitrogen.mathforum.org
in alt.math.undergrad:
Thank you for your answers.
The "Nested Intervals Theorem" states that the
intersection of a nested sequence of intervals in R will
never be empty.
No, it doesn't: it says that the intersection of a nested
sequence of non-empty closed intervals in R is not empty.
You'll need those to be bounded.
WHY?
BECAUSE OTHERWISE THE STATEMENT IS FALSE.
Say I_n is the closed interval [n, infinity). What's the intersection
of the I_n?
David C. Ullrich
|
I thought of that about 30 seconds after posting. |
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Brian M. Scott
Guest
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| Posted: Fri Jun 20, 2008 1:52 am Post subject: Re: Nested sequence of intervals and the empty set |
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On Wed, 18 Jun 2008 23:02:58 -0700, The World Wide Wade
<aderamey.addw@comcast.net> wrote in
<news:aderamey.addw-832CE7.23025818062008@newsgroups.comcast.net>
in alt.math.undergrad:
| Quote: |
In article <1e2g7oyhx3ikm.sfc98i1qz8cz.dlg@40tude.net>,
"Brian M. Scott" <b.scott@csuohio.edu> wrote:
On Mon, 16 Jun 2008 01:56:20 EDT, Guillaume Filteau
filteau@unc.edu> wrote in
news:17423020.1213595810828.JavaMail.jakarta@nitrogen.mathforum.org
in alt.math.undergrad:
Thank you for your answers.
The "Nested Intervals Theorem" states that the
intersection of a nested sequence of intervals in R will
never be empty.
No, it doesn't: it says that the intersection of a nested
sequence of non-empty closed intervals in R is not empty.
You'll need those to be bounded.
|
True.
Brian |
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Guillaume Filteau
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| Posted: Mon Jul 14, 2008 8:03 pm Post subject: Re: Nested sequence of intervals and the empty set |
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| Hello Virgil, could you please remove my address from your message? Thanks ! |
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