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)?
> 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.
Note that it does not follow in general that the limit is unique.
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. Is there another work-around?
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. Is there another work-around?
What is your definition of "topological space"? I would think that "closed" will then be easy to define.
> 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.
On Fri, 4 Jul 2008 06:41:24 -0700 (PDT), Edward Green
<spamspamsp...@netzero.com> 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)?
One desn't use sequences --- look at "nets" --a generalization of sequences. And, "filters".
<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.
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.)
> 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?
> 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.
It does not. For Cauchy convergence or uniform convergence addition 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?
<spamspamsp...@netzero.com> wrote: >On Jul 5, 10:16 am, David C. Ullrich <dullr...@sprynet.com> wrote: >> 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.
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.
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.)
