Convergence in topological spaces
Edward Green
without additional structure on the space? If so, how, since we can't
use the concept of distance (a metric)?
Jannick Asmus
The sequence (x_n) in a topological space X converges to the point x in
X iff (by definition), for every open subset U containing x, U contains
all but finitely many x_n's.
Note that it does not follow in general that the limit is unique.
HTH.
Best wishes,
J.
Edward Green
> On 04.07.2008 15:41, Edward Green wrote:
>
> > Is the convergence of a sequence defined in a topological space
> > without additional structure on the space? If so, how, since we can't
> > use the concept of distance (a metric)?
>
> The sequence (x_n) in a topological space X converges to the point x in
> X iff (by definition), for every open subset U containing x, U contains
> all but finitely many x_n's.
Aha! Very clever.
> Note that it does not follow in general that the limit is unique.
>
> HTH.
Thanks.
I notice this definition is not going to help with "closed", since we
have no way of saying that a sequence converges if it does not
converge to a point in the space. Is there another work-around?
G. A. Edgar
<74357f80-f692-4671...@m44g2000hsc.googlegroups.com>,
Edward Green <spamsp...@netzero.com> wrote:
What is your definition of "topological space"? I would think that
"closed" will then be easy to define.
In a topological space, "closed" cannot in general be defined using
sequences ... instead one can use "generalized sequences" or "nets" for
that.
http://en.wikipedia.org/wiki/Net_%28mathematics%29
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Narcoleptic Insomniac
> Is the convergence of a sequence defined in a
> topological space without additional structure on the
> space?
Yes, (as others have mentioned)...
> If so, how, since we can't use the concept of distance
> (a metric)?
One common definition is the following
DEFINITION. Let T be a topological space and {x_n} be a
sequence in T. Given L in T, we say that
lim_{n --> oo} x_n = L
..iff, for every neighborhood U of L, there exists a
natural number N such that x_n is in U for all n > N.
The reason that I wanted to reply was that this question
reminded me of a "proof" that one of my professors gave
during the first week of a topology course. You'll either
think it's really cute or completely ridiculous
(or both!) ^_^
PROPOSITION. Let R be the set of real numbers, and
T = {{}, R} be the trivial topology on R (i.e. the only
neighborhoods/open sets are the empty set and R itself).
Let {x_n} be *any* sequence in R, and let L be *any* real
number, then lim_{n --> oo} x_n = L.
PROOF. Clearly the only neighborhood of L is U = R. It
follows that for any N, x_n is in U for all n > N, and
thus lim_{n --> oo} x_n = L. []
It's cute because it illustrates the definition of
convergence in topological spaces, and because you can
make any sequence converge to anything (for example, the
sequence, {x_n = (-1)^n} converges to sqrt(2)).
On the other hand, it's ridiculous because you can make
any sequence converge to anything. The world is a boring
place when limits are not unique.
Regards,
Kyle Czarnecki
Pfs...@aol.com
One desn't use sequences --- look at "nets" --a generalization of
sequences. And, "filters".
David C. Ullrich
I don't follow that at all - the definition of "closed" in a metric
space doesn't require that a sequence converge to anything
other than a point in the space! If E is a subset of a metric
space X then E is closed if whenever (x_n) is a sequence in
E and x_n -> x in X then x is in E.
That's the definition in a metric space and it makes just
as much sense in a general topological space. But you need
to note that it's not a _correct_ definition of _closed_
in a general topological apace. Not because it doesn't
make sense, it's simply not right. You could say that
if E is a subset of a topological space X then E is
_sequentially closed_ if whenever (x_n) is a sequence
in E and x_n -> x in X then x is in E.
But a sequentially closed set need not be closed.
You _can_ give a correct definition in terms of
"nets", which are a generalization of sequences.
>Is there another work-around?
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Edward Green
> On Fri, 4 Jul 2008 07:13:57 -0700 (PDT), Edward Green
> <spamspamsp...@netzero.com> wrote:
> >On Jul 4, 9:50 am, Jannick Asmus <jannick.n...@web.de> wrote:
> >> On 04.07.2008 15:41, Edward Green wrote:
>
> >> > Is the convergence of a sequence defined in a topological space
> >> > without additional structure on the space? If so, how, since we can't
> >> > use the concept of distance (a metric)?
>
> >> The sequence (x_n) in a topological space X converges to the point x in
> >> X iff (by definition), for every open subset U containing x, U contains
> >> all but finitely many x_n's.
>
> >Aha! Very clever.
>
> >> Note that it does not follow in general that the limit is unique.
>
> >> HTH.
>
> >Thanks.
>
> >I notice this definition is not going to help with "closed", since we
> >have no way of saying that a sequence converges if it does not
> >converge to a point in the space.
>
> I don't follow that at all - the definition of "closed" in a metric
> space doesn't require that a sequence converge to anything
> other than a point in the space! If E is a subset of a metric
> space X then E is closed if whenever (x_n) is a sequence in
> E and x_n -> x in X then x is in E.
What I was alluding to was the idea that a sequence could be
convergent in some sense, without necessarily converging to something
in particular -- though we sometimes say in these cases that the
sequence converges to a point not in the space. If there are no such
sequences, then the space is closed.
IIRC such convergence without a necessary limit point is called
"Cauchy", and I was remarking that, following the comment of Jannick
Asmus, I had a notion of convergence in topological spaces, but no
parallel notion of "Cauchy convergence" in such spaces -- assuming one
existed.
Inspired by the remarks here, I offer a candidate:
(Cauchy convergence in a topological space)
If, for a sequence x_n in a topological space T, there exist for all
N
open sets S_N such that x_n, n < N are excluded from S_N, and all
x_n, n >= N are included, then that sequence is (Cauchy) convergent.
Maybe I need to add something about the S_N forming a sequence of
proper sub-sets?
> That's the definition in a metric space and it makes just
> as much sense in a general topological space.
But if the alleged limit point x is not in the space, it seems to me
we don't necessarily know that the original statement means anything:
we are apparently assuming a larger structure in which the target
space is embedded rather than just working from concepts within the
space?
> But you need
> to note that it's not a _correct_ definition of _closed_
> in a general topological apace. Not because it doesn't
> make sense, it's simply not right. You could say that
> if E is a subset of a topological space X then E is
> _sequentially closed_ if whenever (x_n) is a sequence
> in E and x_n -> x in X then x is in E.
>
> But a sequentially closed set need not be closed.
> You _can_ give a correct definition in terms of
> "nets", which are a generalization of sequences.
Oh... well, that's different.
BTW, in case anybody was wondering about my motivation, I am still
trying to autodidact myself on Lie groups using the Dover reprint of
Robert Gilmore's book on the subject, and on page 60, just after he
repeats the basic axioms of a topological space, he remarks that he
will now develop the concepts of "compactness", "closure" and
"continuity", immediately writing of convergent sequences, and leaving
the neophyte wondering if one should blithly assume these ideas work
in a topological space.
Do I have to make a detour through "nets" to proceed?
William Elliot
>
> What I was alluding to was the idea that a sequence could be
> convergent in some sense, without necessarily converging to something
> in particular -- though we sometimes say in these cases that the
> sequence converges to a point not in the space. If there are no such
> sequences, then the space is closed.
>
> IIRC such convergence without a necessary limit point is called
> "Cauchy", and I was remarking that, following the comment of Jannick
> Asmus, I had a notion of convergence in topological spaces, but no
> parallel notion of "Cauchy convergence" in such spaces -- assuming one
> existed.
>
structure is used. Those spaces are called uniform spaces. The classic
example of uniform spaces are metric spaces. Another example of uniform
spaces are topological groups. The essence of uniform spaces is that the
size or smallness of an open nhood of of a point an be preserved or
maintained for all points.
For example, in metric spaces, B(x,r) is a small nhood
of x which is equally small as B(a,r) for any point a.
> Inspired by the remarks here, I offer a candidate:
>
> (Cauchy convergence in a topological space)
>
> If, for a sequence x_n in a topological space T, there exist for all
> N
> open sets S_N such that x_n, n < N are excluded from S_N, and all
> x_n, n >= N are included, then that sequence is (Cauchy) convergent.
>
> Maybe I need to add something about the S_N forming a sequence of
> proper sub-sets?
>
A space is first countable when every point has a countable base of open
sets. These open sets can be arranged in subset descending order.
However for Cauchy sequences, one doesn't look to see how close the
sequence is coming to a point of convergence. One compares two points
to see how close together they are.
I suppose, some descending sequence of open sets (Uj)_j
with finite /\_j Uj or |/\_j Uj| <= 1. Then (xj)_j is a Cauchy
sequence when there exists such a sequence of open sets and
for all j, some n with for all r,s > n, x_r, x_s in Uj.
The immediate problem is Uj = (j,oo) and x_j = j.
Hm, then make |/\_j Uj| = 1. Well no, then
a sequence of rations converging to pi, would within the space
of nationals not be a Cauchy sequence even though it doesn't
converge within it's space of nationals.
> > That's the definition in a metric space and it makes just
> > as much sense in a general topological space.
>
It does not. You've many details to work out and when you do, I'll bet
you'll come to results of past mathematicians who first consider this.
Namely that of a uniform space, which I remind you, metric spaces are.
> But if the alleged limit point x is not in the space, it seems to me
> we don't necessarily know that the original statement means anything:
> we are apparently assuming a larger structure in which the target
> space is embedded rather than just working from concepts within the
> space?
>
Yes. Uniform space. See the article on uniform spaces in Wikipedia.
> > But you need to note that it's not a _correct_ definition of _closed_
> > in a general topological apace. Not because it doesn't make sense,
> > it's simply not right. You could say that if E is a subset of a
> > topological space X then E is _sequentially closed_ if whenever (x_n)
> > is a sequence in E and x_n -> x in X then x is in E.
When a set is closed, every convergent sequence within the set, converges
to a point within the set. There's a converse, but only for 1st countable
spaces (which metric spaces are). When a space is 1st countable, then
if a set has the property that every converging sequence within the set
converges to a point in the set, then the set is closed. For other
spaces, such as omega_1 + 1 (an example why 1st countable is needed)
sequences have been generalized to the notion of net where closed sets can
be described by nets, like closed sets within 1st countable spaces can be
described by sequences. Nets are an America invention. Filters, which
were invented during the same time and for similar reasons than nets,
were invented in Europe, France IIRC.
> > But a sequentially closed set need not be closed.
> > You _can_ give a correct definition in terms of
> > "nets", which are a generalization of sequences.
>
> Oh... well, that's different.
>
> BTW, in case anybody was wondering about my motivation, I am still
> trying to autodidact myself on Lie groups using the Dover reprint of
> Robert Gilmore's book on the subject, and on page 60, just after he
> repeats the basic axioms of a topological space, he remarks that he
> will now develop the concepts of "compactness", "closure" and
> "continuity", immediately writing of convergent sequences, and leaving
> the neophyte wondering if one should blithely assume these ideas work
> in a topological space.
>
> Do I have to make a detour through "nets" to proceed?
>
Detour? Uniform spaces, nets and filters (all described in Wikipedia)
aren't detours. They're back ground material.
What does 'autodidact' mean?
--
Riddle of the day. Is the current administration a lie group?
David C. Ullrich
No, we most definitely do not say that. There's no such thing
as a closed topological space or a closed metric space. We
speak of closed _subsets_ of a topological space or of a
metric space. (A set can be closed when regarded as a subset
of one space and not closed when regarded as a subset
of another, for example.)
>IIRC such convergence without a necessary limit point is called
>"Cauchy",
There certainly is such a thing as a Cauchy sequence (in a metric
space). A Cauchy sequence is _not_ a convergent sequence.
(Thinking about it as "convergent, but not converging to
anything in particular" might not be a bad idea to understand
what the concept means. Or it might be a bad idea - in any
case it's not _correct_ to say that.)
You're confusing "closed" and "complete". A metric space
is _complete_ if every Cauchy sequence converges.
(Possibly a reason for the confusion is that if X is
a complete metric space and E is a subset of X then
E is closed in X if and only if E, regarded as a metric
space in itself, is complete. But that only applies to
subsets of _complete_ spaces.)
>and I was remarking that, following the comment of Jannick
>Asmus, I had a notion of convergence in topological spaces, but no
>parallel notion of "Cauchy convergence" in such spaces -- assuming one
>existed.
There _is_ a notion of convergence in general topological
spaces, and there is no notion of "Cauchy sequence".
>Inspired by the remarks here, I offer a candidate:
>
>(Cauchy convergence in a topological space)
>
> If, for a sequence x_n in a topological space T, there exist for all
>N
> open sets S_N such that x_n, n < N are excluded from S_N, and all
> x_n, n >= N are included, then that sequence is (Cauchy) convergent.
??? This is not equivalent to the usual definition in metric spaces.
>Maybe I need to add something about the S_N forming a sequence of
>proper sub-sets?
>
>> That's the definition in a metric space and it makes just
>> as much sense in a general topological space.
>
>But if the alleged limit point x is not in the space,
If we're talking about a topological space X and a
convergent sequence in X then the limit _is_ in X.
>it seems to me
>we don't necessarily know that the original statement means anything:
>we are apparently assuming a larger structure in which the target
>space is embedded rather than just working from concepts within the
>space?
No, I just didn't state explicitly that x was an element of X above,
thinking that was understood.
>> But you need
>> to note that it's not a _correct_ definition of _closed_
>> in a general topological apace. Not because it doesn't
>> make sense, it's simply not right. You could say that
>> if E is a subset of a topological space X then E is
>> _sequentially closed_ if whenever (x_n) is a sequence
>> in E and x_n -> x in X then x is in E.
>>
>> But a sequentially closed set need not be closed.
>> You _can_ give a correct definition in terms of
>> "nets", which are a generalization of sequences.
>
>Oh... well, that's different.
>
>BTW, in case anybody was wondering about my motivation, I am still
>trying to autodidact myself on Lie groups using the Dover reprint of
>Robert Gilmore's book on the subject, and on page 60, just after he
>repeats the basic axioms of a topological space, he remarks that he
>will now develop the concepts of "compactness", "closure" and
>"continuity", immediately writing of convergent sequences, and leaving
>the neophyte wondering if one should blithly assume these ideas work
>in a topological space.
>
>Do I have to make a detour through "nets" to proceed?
Probably all the spaces under consideration are metric
spaces.
Edward Green
<...>
> What does 'autodidact' mean?
"Self taught", although I made it into a verb, to teach myself.
Such a richness of replies! So little time!
I must mull carefully.
Edward Green
<...>
> > But if the alleged limit point x is not in the space, it seems to me
> > we don't necessarily know that the original statement means anything:
> > we are apparently assuming a larger structure in which the target
> > space is embedded rather than just working from concepts within the
> > space?
>
> Yes. Uniform space. See the article on uniform spaces in Wikipedia.
Aha! That immediately cleared some things up:
From the Wikipedia article:
"Uniform spaces generalize metric spaces and topological groups..."
Since topological groups are where Gilmore is headed, perhaps he
tacitly assumed this extra structure after introducing topological
spaces, just before he continued with the definitions involving limits
which aroused my suspicions.
This illustrates a general principle I noted a long time ago: it is
practically impossible for an experienced practitioner to write a
fully coherent introduction to a subject, i.e. one which builds only
on what he explicitly states and not on what he knows ahead of time,
though the reader does not, to be the finished structure.
> > > But you need to note that it's not a _correct_ definition of _closed_
> > > in a general topological apace. Not because it doesn't make sense,
> > > it's simply not right. You could say that if E is a subset of a
> > > topological space X then E is _sequentially closed_ if whenever (x_n)
> > > is a sequence in E and x_n -> x in X then x is in E.
>
> When a set is closed, every convergent sequence within the set, converges
> to a point within the set. There's a converse, but only for 1st countable
> spaces (which metric spaces are). When a space is 1st countable, then
> if a set has the property that every converging sequence within the set
> converges to a point in the set, then the set is closed. For other
> spaces, such as omega_1 + 1 (an example why 1st countable is needed)
> sequences have been generalized to the notion of net where closed sets can
> be described by nets, like closed sets within 1st countable spaces can be
> described by sequences. Nets are an America invention. Filters, which
> were invented during the same time and for similar reasons than nets,
> were invented in Europe, France IIRC.
I noticed that in this thread, about equal numbers of people said in
effect, yes, you can generalize sequences to topological spaces, while
others said, no, what you need is nets!
I am approaching the subject with the goals of a thwarted physicist
rather than a thwarted mathematician, and I wonder if, from the point
of view of understanding how physicists use Lie groups -- for getting
a "physicist's understanding" of Lie groups, one gets by without some
detours/background.
<...>
Edward Green
> On Sat, 5 Jul 2008 09:04:04 -0700 (PDT), Edward Green
> >What I was alluding to was the idea that a sequence could be
> >convergent in some sense, without necessarily converging to something
> >in particular -- though we sometimes say in these cases that the
> >sequence converges to a point not in the space. If there are no such
> >sequences, then the space is closed.
>
> No, we most definitely do not say that. There's no such thing
> as a closed topological space or a closed metric space.
> We speak of closed _subsets_ of a topological space or of a
> metric space. (A set can be closed when regarded as a subset
> of one space and not closed when regarded as a subset
> of another, for example.)
Gilmore (op. cit., p. 59-60)
"A topological space T is a set of points on which...", and then:
"A space T is compact if ... ", and then:
"A set T is closed if it contains all its limit points".
Hmm... It is unfortunate that, although he now calls it a set and not
a space, Gilmore just got through using T to represent the set
underlying an entire topological space. You can see the reason for my
confusion.
Perhaps this is an editing error.
> >IIRC such convergence without a necessary limit point is called
> >"Cauchy",
>
> There certainly is such a thing as a Cauchy sequence (in a metric
> space). A Cauchy sequence is _not_ a convergent sequence.
> (Thinking about it as "convergent, but not converging to
> anything in particular" might not be a bad idea to understand
> what the concept means. Or it might be a bad idea - in any
> case it's not _correct_ to say that.)
>
> You're confusing "closed" and "complete".
See above.
<...>
> >Do I have to make a detour through "nets" to proceed?
>
> Probably all the spaces under consideration are metric
> spaces.
That would simplify things, wouldn't it: I wouldn't even have to worry
about uniform spaces, which I see are a "generalization" of metric
spaces! Maybe that's the trick: the author starts in broad
topological generality, but in the end everthing is a metric space.
Dave Seaman
> I noticed that in this thread, about equal numbers of people said in
> effect, yes, you can generalize sequences to topological spaces, while
> others said, no, what you need is nets!
Wrong. Not one person has said that you can't generalize sequences to
topological spaces. That's nonsense.
What people are telling you is that in a general topological space, there
is a difference between "closed" and "sequentially closed". That is,
sequences exist, but there are some situations for which they are not a
sufficiently powerful tool.
> I am approaching the subject with the goals of a thwarted physicist
> rather than a thwarted mathematician, and I wonder if, from the point
> of view of understanding how physicists use Lie groups -- for getting
> a "physicist's understanding" of Lie groups, one gets by without some
> detours/background.
><...>
--
Dave Seaman
Third Circuit ignores precedent in Mumia Abu-Jamal ruling.
<http://www.indybay.org/newsitems/2008/03/29/18489281.php>
Dave L. Renfro
>> I noticed that in this thread, about equal numbers of people
>> said in effect, yes, you can generalize sequences to topological
>> spaces, while others said, no, what you need is nets!
Dave Seaman wrote (in part):
> Wrong. Not one person has said that you can't generalize sequences
> to topological spaces. That's nonsense.
>
> What people are telling you is that in a general topological space,
> there is a difference between "closed" and "sequentially closed".
> That is, sequences exist, but there are some situations for which
> they are not a sufficiently powerful tool.
To elaborate on what Dave Seaman wrote, the closure of a set
is the intersection of all closed sets containing the set.
It is not difficult to show that this process, taking the
intersection of all closed sets containing a given set,
produces a closed set (meaning that its complement is an
open set in the topology under consideration). Moreover,
the closed set produced is the smallest (with respect to
set inclusion) closed set that contains the given set.
[Note: It's not automatic that collections of objects
contain a smallest object -- there is no smallest positive
real number and, in the real line, there is no smallest open
set containing the closed interval [0,1]. However, any nonempty
collection of closed sets contains a smallest set -- just take
the intersection of all the closed sets in the collection.]
In the case of sequences, if you take all sequences of points
in a given set and adjoin (add to, include, form the union with)
to the given set all the limits of those sequences that happen
to be convergent in the underlying topological space (some of
these points will belong to the given set already, as you can
see by considering constant sequences), then you get what is
called the "sequential closure of the set". The sequential
closure of a set has the property that it is "sequentially
closed", meaning that if you take a sequence of points in the
sequential closure that converges in the underlying topological
space, then the limit of this sequence belongs to the sequential
closure. This follows from a result that can be roughly stated
as "a limit of limits is a limit". It can be proved that the
sequential closure of a set is always a subset of the closure
of the set. However, in a topological space (but not a metric
space or, more generally, in a space that has a countable base
at each point), it is possible that the sequential closure
is a proper subset of the closure. What happens is that in
some topological spaces there are so many closed sets (resulting
from there being a lot of open sets), at least so many closed
sets having certain relationships with other sets, that limits
of sequences "don't reach" all the points necessary to form the
closure of the set. It's as if you're shooting bullets at a
house that has some bullet-proof windows and some windows that
are not bullet-proof. The bullet-proof windows will be left
unbroken. In the case of looking at limits of convergent sequences
(again, convergent meaning the sequence converges to some point
in the space, a notion that is independent of what set we're
looking at, at least once we have the sequence in hand), we
get some of the points needed to form the closure of the set,
but not necessarily all of them.
There is a notion of transfinite sequences, where you allow
the sequences to "travel further", even have uncountable
lengths (instead using functions from the set of positive
integers into the space, which is what sequences are, use
functions from any nonzero limit ordinal into the space).
One might think that this would do the trick, but it doesn't.
Although using transfinite sequences will give you a possibly
larger closure notion, there are topological spaces in which
even arbitrary transfinite-length sequences are not sufficient
to capture every point in the (usual) closure of a set. For
a simple example of this last situation, see E. S. Wolk,
"On the inadequacy of cofinal subnets and transfinite
sequences", American Mathematical Monthly 89 (1982), 310-311.
Dave L. Renfro
Dave L. Renfro
> is a proper subset of the closure. What happens is that in
> some topological spaces there are so many closed sets (resulting
> from there being a lot of open sets), at least so many closed
> sets having certain relationships with other sets, that limits
> of sequences "don't reach" all the points necessary to form the
> closure of the set.
I guess this should be "so few closed sets having certain ..."!
Since the closure is the intersection of all closed sets such
that ..., larger closures would arise from having fewer closed
sets around to use.
Dave L. Renfro
David C. Ullrich
<spamsp...@netzero.com> wrote:
>On Jul 6, 9:11 am, David C. Ullrich <dullr...@sprynet.com> wrote:
>> On Sat, 5 Jul 2008 09:04:04 -0700 (PDT), Edward Green
>
>> >What I was alluding to was the idea that a sequence could be
>> >convergent in some sense, without necessarily converging to something
>> >in particular -- though we sometimes say in these cases that the
>> >sequence converges to a point not in the space. If there are no such
>> >sequences, then the space is closed.
>>
>> No, we most definitely do not say that. There's no such thing
>> as a closed topological space or a closed metric space.
>
>> We speak of closed _subsets_ of a topological space or of a
>> metric space. (A set can be closed when regarded as a subset
>> of one space and not closed when regarded as a subset
>> of another, for example.)
>
>Gilmore (op. cit., p. 59-60)
>
>"A topological space T is a set of points on which...", and then:
>
>"A space T is compact if ... ", and then:
>
>"A set T is closed if it contains all its limit points".
It's impossible to tell whether these are correct since
you leave so much out.
>Hmm... It is unfortunate that, although he now calls it a set and not
>a space, Gilmore just got through using T to represent the set
>underlying an entire topological space. You can see the reason for my
>confusion.
I can?
>Perhaps this is an editing error.
There's nothing erroneous about the above that I can see.
A topological space _is_ a set (more formally, a topological
space is an ordered pair (X, tau) where X is a set and tau
is a topology on X. But people usually refer to X itself
as a topological space). A topological space is a set;
it does not follow that any time he says "set" he means
"topological space"! The error is careless reading.
>> >IIRC such convergence without a necessary limit point is called
>> >"Cauchy",
>>
>> There certainly is such a thing as a Cauchy sequence (in a metric
>> space). A Cauchy sequence is _not_ a convergent sequence.
>> (Thinking about it as "convergent, but not converging to
>> anything in particular" might not be a bad idea to understand
>> what the concept means. Or it might be a bad idea - in any
>> case it's not _correct_ to say that.)
>>
>> You're confusing "closed" and "complete".
>
>See above.
>
><...>
>
>> >Do I have to make a detour through "nets" to proceed?
>>
>> Probably all the spaces under consideration are metric
>> spaces.
>
>That would simplify things, wouldn't it: I wouldn't even have to worry
>about uniform spaces, which I see are a "generalization" of metric
>spaces! Maybe that's the trick: the author starts in broad
>topological generality, but in the end everthing is a metric space.
As I said, it's impossible to tell whether he makes any errors
regarding what we can or cannot do with sequences from
what you quote above, since you omit so much.
Edward Green
> On Sun, 6 Jul 2008 19:44:09 -0700 (PDT), Edward Green
>
>
>
>
>
> <spamspamsp...@netzero.com> wrote:
> >On Jul 6, 9:11 am, David C. Ullrich <dullr...@sprynet.com> wrote:
> >> On Sat, 5 Jul 2008 09:04:04 -0700 (PDT), Edward Green
>
> >> >What I was alluding to was the idea that a sequence could be
> >> >convergent in some sense, without necessarily converging to something
> >> >in particular -- though we sometimes say in these cases that the
> >> >sequence converges to a point not in the space. If there are no such
> >> >sequences, then the space is closed.
>
> >> No, we most definitely do not say that. There's no such thing
> >> as a closed topological space or a closed metric space.
>
> >> We speak of closed _subsets_ of a topological space or of a
> >> metric space. (A set can be closed when regarded as a subset
> >> of one space and not closed when regarded as a subset
> >> of another, for example.)
>
> >Gilmore (op. cit., p. 59-60)
>
> >"A topological space T is a set of points on which...", and then:
>
> >"A space T is compact if ... ", and then:
>
> >"A set T is closed if it contains all its limit points".
>
> It's impossible to tell whether these are correct since
> you leave so much out.
I wasn't commenting on whether they were "correct" or not. I was
merely pointing out how the author sigued from using "T" to mean the
underlying set of points of a topological space, to, presumably, the
underlying set plus its structure, by which he labels it a "space", to
refering to some _other_ set by the same letter, presumably no longer
the original underlying set of the topological space but a subset.
> >Hmm... It is unfortunate that, although he now calls it a set and not
> >a space, Gilmore just got through using T to represent the set
> >underlying an entire topological space. You can see the reason for my
> >confusion.
>
> I can?
>
> >Perhaps this is an editing error.
>
> There's nothing erroneous about the above that I can see.
Editing errors include items not merely 100% logically wrong, but
misleading.
> A topological space _is_ a set (more formally, a topological
> space is an ordered pair (X, tau) where X is a set and tau
> is a topology on X. But people usually refer to X itself
> as a topological space). A topological space is a set;
> it does not follow that any time he says "set" he means
> "topological space"! The error is careless reading.
Using "T" twice in quick succession to mean the set underlying a
topological space and then immediately again to mean just some old set
which is now not the underlying set of a topological space, when any
other letter, such as S, would have served better, is plain old sloppy
editing, or simply perverse. This recurrence wasn't 15 pages latter,
but immediate.
Unless Gilmore happens to be your uncle, I'm not sure why you are so
quick to resort to the typical antagonistic Usenetism "No... it is
_you_ that are a sloppy reader!". It wasn't even your writing, and
you don't have the sample in front of you. What makes you so sure?
I will forgo further exchanges here in the interest of my time and
blood pressure. Too many words on too little. I wish to learn some
math, not defend myself against pointless criticism.
Edward Green
> On Sun, 6 Jul 2008 19:17:55 -0700 (PDT), Edward Green wrote:
> > I noticed that in this thread, about equal numbers of people said in
> > effect, yes, you can generalize sequences to topological spaces, while
> > others said, no, what you need is nets!
>
> Wrong.
Thank you, Dr. Pauli.
> Not one person has said that you can't generalize sequences to
> topological spaces.
If you want to be legalistic about it, which I guess you do, I didn't
say that either. I implied that's not what one needs, as I think did:
On Jul 4, 10:59 am, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
wrote:
> In a topological space, "closed" cannot in general be defined using
> sequences ... instead one can use "generalized sequences" or "nets" for
> that.
On Jul 5, 9:34 am, Pfss...@aol.com wrote:
> One desn't use sequences --- look at "nets" --a generalization of
> sequences. And, "filters".
Hmm... I see now that I incautiously used the word "generalize"! You
could generalize sequences in the sense of having something which
looked like, and was called, a "sequence" in a topological space,
which is what I had in mind when I asked my naive question, or you can
have some kind of "generalized sequence", which (presumably) doesn't
look exactly like a sequence, and gets a brand new name.
It does indeed seem possible to "generalize sequences" in the first
sense to topological spaces, as Jannick Asmus implied:
On Jul 4, 9:50 am, Jannick Asmus <jannick.n...@web.de> wrote:
> The sequence (x_n) in a topological space X converges to the point x in
> X iff (by definition), for every open subset U containing x, U contains
> all but finitely many x_n's.
and Kyle Czarnecki
On Jul 4, 11:41 am, Narcoleptic Insomniac <wrote>:
> DEFINITION. Let T be a topological space and {x_n} be a
> sequence in T. Given L in T, we say that
>
> lim_{n --> oo} x_n = L
>
> ..iff, for every neighborhood U of L, there exists a
> natural number N such that x_n is in U for all n > N.
but I have the idea these generalizations of sequences are not the
"generalized sequences" referred to under the rubric "net" or
"filter".
But you know what, Dave? If you want to prove I'm stupid you probably
won't have too much difficulty, because no doubt I'm very stupid. So
I hope the less antagonistic contributors will not be offended very
much if I cease communications with you and that fellow David, since,
as I said, by showing I'm stupid you are probably beating a dead
horse, although the parallel implications of malfeasance do rather get
under my skin.
Dave Seaman
> On Jul 6, 11:27 pm, Dave Seaman <dsea...@no.such.host> wrote:
>> On Sun, 6 Jul 2008 19:17:55 -0700 (PDT), Edward Green wrote:
>> > I noticed that in this thread, about equal numbers of people said in
>> > effect, yes, you can generalize sequences to topological spaces, while
>> > others said, no, what you need is nets!
>>
>> Wrong.
> Thank you, Dr. Pauli.
>> Not one person has said that you can't generalize sequences to
>> topological spaces.
> If you want to be legalistic about it, which I guess you do, I didn't
> say that either. I implied that's not what one needs, as I think did:
No, you said that others said it. That's the claim that I objected to.
> On Jul 4, 10:59 am, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
> wrote:
>> In a topological space, "closed" cannot in general be defined using
>> sequences ... instead one can use "generalized sequences" or "nets" for
>> that.
That's a statement about topology, not a statement about sequences.
> On Jul 5, 9:34 am, Pfss...@aol.com wrote:
>> One desn't use sequences --- look at "nets" --a generalization of
>> sequences. And, "filters".
That is, one doesn't use sequences to define closed sets. That doesn't
mean that one mustn't ever use sequences on topological spaces.
> Hmm... I see now that I incautiously used the word "generalize"! You
> could generalize sequences in the sense of having something which
> looked like, and was called, a "sequence" in a topological space,
> which is what I had in mind when I asked my naive question, or you can
> have some kind of "generalized sequence", which (presumably) doesn't
> look exactly like a sequence, and gets a brand new name.
A sequence on a set X, by definition, is a mapping f: N -> X, where N is
the natural numbers. Therefore, talking about a "generalization" of
sequences to topological spaces makes no sense, because the same
definition holds whether X happens to have a topology defined on it or
not. Sequences on topological spaces are a special case, not a
generalization, of sequences on sets.
You need a topology before you can talk about convergence, of course, but
convergence is not a part of the definition of a sequence.
> It does indeed seem possible to "generalize sequences" in the first
> sense to topological spaces, as Jannick Asmus implied:
See above. A special case is the very opposite of a generalization.
> On Jul 4, 9:50 am, Jannick Asmus <jannick.n...@web.de> wrote:
>> The sequence (x_n) in a topological space X converges to the point x in
>> X iff (by definition), for every open subset U containing x, U contains
>> all but finitely many x_n's.
That's a special case of convergence, assuming you are coming from the context
of convergence in a metric space. It is not a special case of a sequence.
> and Kyle Czarnecki
> On Jul 4, 11:41 am, Narcoleptic Insomniac <wrote>:
>> DEFINITION. Let T be a topological space and {x_n} be a
>> sequence in T. Given L in T, we say that
>> lim_{n --> oo} x_n = L
>> ..iff, for every neighborhood U of L, there exists a
>> natural number N such that x_n is in U for all n > N.
> but I have the idea these generalizations of sequences are not the
> "generalized sequences" referred to under the rubric "net" or
> "filter".
They are not generalizations of sequences. They are a generalization of
the notion of convergence in metric spaces. But you are correct that the
definitions of nets and filters, and of convergence thereof, are yet more
general.
> But you know what, Dave? If you want to prove I'm stupid you probably
> won't have too much difficulty, because no doubt I'm very stupid. So
> I hope the less antagonistic contributors will not be offended very
> much if I cease communications with you and that fellow David, since,
> as I said, by showing I'm stupid you are probably beating a dead
> horse, although the parallel implications of malfeasance do rather get
> under my skin.
--
Dave Seaman
>> On Jul 4, 9:50 am, Jannick Asmus <jannick.n...@web.de> wrote:
>>> The sequence (x_n) in a topological space X converges to the point x in
>>> X iff (by definition), for every open subset U containing x, U contains
>>> all but finitely many x_n's.
> That's a special case of convergence, assuming you are coming from the context
> of convergence in a metric space. It is not a special case of a sequence.
I meant, that's a generalization of convergence, not a generalization of
a sequence.
David C. Ullrich
Oh. I'm curious whether what's in the book is correct or not.
At the start of this you seemed to think that, for example,
closed sets in a topological space could be characterized
by sequences, you seemed to be saying that you'd got this
mistaken impression from that book, and then it becomes
a natural question whether it really says that in the book or not.
Because you claim to be interested in learning the math, it seems
likely that the problems here were caused by your sloppy reading,
and if you really _are_ interested in learning the math you'd
be better off realizing this, so you'd read more carefully in the
future.
>It wasn't even your writing, and
>you don't have the sample in front of you. What makes you so sure?
I'm not sure - as I point out above, I can't tell from the quotes
you've given whether what the book says is correct or not.
But _you_ seemed to be offering those quotes as evidence of
errors in the book, or at least as evidence that the writing in
the book is not clear. Whether or not the book is corect or
clear, those quote are _not_ evidence of any such thing.
I can be sure of that because I've seen the quotes I'm talking about.
>I will forgo further exchanges here in the interest of my time and
>blood pressure. Too many words on too little. I wish to learn some
>math, not defend myself against pointless criticism.
If you say so. It doesn't _seem_ as though learning the math is
your primary objective, it seems as though what you really want
to do is explain why your misconceptions were due to what it
says in that book.