Talk:Topological group

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Continuous group[edit]

Is it OK to say that 'Continuous group' is the same than 'Topological group' ??

looxix 23:30 Feb 23, 2003 (UTC)

Continuous group is an older term that possibly could also mean Lie group, so I would be careful. AxelBoldt 21:16 Mar 2, 2003 (UTC)

Left uniformity[edit]

Although it is true that the left uniformity turns left multiplications into uniformly continuous maps, this is not a characterization of the left uniformity. In fact, the right uniformity also turns left multiplications into uniformly continuous maps. The important characteristic of the left uniformity is that it is invariant with respect to left translations. This is not mentioned on the page. —Preceding unsigned comment added by 65.95.217.43 (talkcontribs) 14:22, 16 April 2006

objects in analysis[edit]

"Almost all objects investigated in analysis are topological groups (usually with some additional structure)."

The above statement, which appears at the beginning of the article, is ambiguous. For example, continuous functions, open sets, and Lebesgue measure are not topological groups in any reasonable sense. Can someone rephrase the sentence? --Acepectif 03:45, 19 June 2007 (UTC)

H-spaces[edit]

After the remark that the fundamental group of a topological group is abelian, this is deduced from the corresponding property for H-spaces. This is silly: the proof for topological groups works the same way and is simpler, so H-spaces introduce needless generality. —Preceding unsigned comment added by 137.82.36.10 (talkcontribs) 21:08, 27 August 2007

I do not fully agree with the logic of this remark. If you have a general theorem it seems to me perfectly reasonable to cite the theorem in justification of a particular case of it, whether or not the particular case can be proved more simply. Apart from anything else this helps to show relationships among different parts of the subject. Nevertheless I have reworded this statement in an attempt to answer this criticism. JamesBWatson (talk) 12:59, 2 December 2008 (UTC)

Closure of compact sets[edit]

"Topology Expert" has asked me to prove that, in a topological group, the closure of a compact set is compact. But this is a well-known property of regular spaces, and topological groups are regular (as already stated in the article). --Zundark (talk) 11:52, 31 October 2008 (UTC)

What you say above is meaningless to the context. You wrote that the Hausdorff condition is not required for the theorem in question to hold since the closure of a compact set in a topological group is compact. But for this to hold, you require a stronger condition than the Hausdorff axiom. So you can't conclude this from the fact that topological groups are regular.

Topology Expert (talk) 04:35, 1 November 2008 (UTC)

You have claimed on my talk page that you are a topologist. If this is true, then you must be well aware that "regular space" is used in two senses, only one of which is stronger than the Hausdorff condition. I even linked the phrase "regular spaces" so that you could go and read the definition if you were uncertain what I meant. Please do that now. All topological groups are uniformizable, and therefore completely regular. They need not be Hausdorff. --Zundark (talk) 10:57, 1 November 2008 (UTC)
I am well aware of this. But if you assume that one-point sets are closed in the topological group you can get complete regularity and the Hausdorff condition from there. Furthermore, in many branches of mathematics, topological groups are required to be T1 (for instance Lie group theory where a stronger assumption exists).

However, I accept that what you said earlier was pertinent to the discussion so that's fine. I'll respond to your next message as soon as I can because I have some comments.

Topology Expert (talk) 11:28, 1 November 2008 (UTC)

There is, in fact, a formula for the closure of a compact set in a topological group: if K is compact, then (which is clearly compact, as it's a continuous image of the compact product space ). Some Googling finds an on-line proof of this formula: Introduction to Topological Groups by Karl Heinrich Hofmann (it's part (iv) of Corollary 1.16, on page 9). Maybe it's worth mentioning this formula in the article (though it's obviously only of interest for non-Hausdorff topological groups). --Zundark (talk) 11:52, 31 October 2008 (UTC)

Why is the closure of {1} compact (I don't have time to verify this myself (not that I can't) so could you please prove it? Until then, I will revert your edits)?

Topology Expert (talk) 09:15, 4 November 2008 (UTC)

Why is your ignorance justification for reverting my edits? How would you like it if I went around reverting your edits on the grounds that I haven't got time to check that they're correct?
The closure of {1} is compact because its topology is trivial. This is obvious, but see part (ii) of that same Corollary 1.16 if you can't work it out yourself. --Zundark (talk) 09:39, 4 November 2008 (UTC)
Wrong again. According to your argument (you say that {1} inherits the trivial topology), the closure of a one-point set in a topological space is always compact (since one-point sets always inherit the trivial topology in case you did not know). What about an infinite set in the particular point topology? The closure of the particular point is not compact (not even metacompact!), yet the topology it inherits is trivial. I recommend that you 'relearn' topology (I doubt that you have learnt it in the first place), if you are going to place personal attacks (why did you link trivial topology?) just to hide your misunderstanding.
Topology Expert (talk) 03:47, 5 November 2008 (UTC)
I have not made any mistake here. You appear to have completely misunderstood what I said. I said that the closure of {1} has the trivial topology. This is true, and I pointed you to a proof. I never claimed that the closure of a singleton in an arbitrary topological space is compact, so your example with the particular point topology is irrelevant. --Zundark (talk) 09:21, 5 November 2008 (UTC)
Claim: In a topological group, the closure of each singleton has the indiscrete topology and is thus compact.
Proof: Define an equivalence relation on the group: iff (where quantification is over the open sets). Doing a little work you can see that this is an equivalence relation. Now, it turns out that the equivalence classes are . Moreover one can see that every open set (indeed every borel set... indeed every set generated from the open sets by boolean set operations) are -invariant. Thus for an open set and the closure of a singleton, or (which one depends on whether or not ). Thus the subspace topology of is, as Zundark said, trivial. Hence it is compact.
drusus null 00:45, 7 April 2010 (UTC)

To resolve this dispute once an for all, see my recent edit. Now, removing the Hausdorff condition will definitely make the statement false (not that I accept that the Hausdorff condition can be ommitted from the previous statement (nor do I accept your above arguments) but at least in this case there will be no dispute).

Topology Expert (talk) 04:30, 5 November 2008 (UTC)

I've provided two different proofs of the part you were stuck on. Your rejection of the second one was based on a misreading of what I wrote, and you haven't yet given a reason for rejecting the first one. If you could be more specific about what is not clear to you, I can provide further explanation or references. --Zundark (talk) 15:42, 5 November 2008 (UTC)

Proposed rewrite[edit]

On 17 September 2008 Topology Expert tagged this page as possibly requiring a complete rewrite. I have read the page and cannot see why this should be. I have also looked at the talk page to see whether Topology Expert gives any explanation there: he doesn't. Topology Expert has disputed a few facts concerning closures of compact sets with Zundark, but a dispute on one point scarcely justifies a complete rewrite. Can anyone justify keeping the tag there? JamesBWatson (talk) 13:01, 2 December 2008 (UTC)

I've removed the tag. The article could obviously be improved, but it's no worse than many others, and the tag doesn't really seem justified. The discussion between me and "Topology Expert" above is not relevant, as "Topology Expert" eventually admitted that I was right anyway. --Zundark (talk) 14:02, 2 December 2008 (UTC)
I am putting the tag back with the following justification:
  • There are many important results that could be included about topological groups such as its applications to harmonic analysis and Lie group theory
  • The article is not structured well; different facts are places in a haphazard manner

Topology Expert (talk) 09:33, 4 December 2008 (UTC)

Under those conditions, wouldn't most articles qualify for this tag? We need some sort of threshold to fall below before we apply the tag, and I think any reasonable threshold would exclude this article. Topology expert, why not make use of the comments page, which is included in the rating template above? This way your ideas are still noted until taken care of, and we aren't in the situation of applying rewrite tags to articles whenever they're not close to good articles. Ben (talk) 23:40, 6 December 2008 (UTC)

I am certainly glad that I raised this question, as Topology Expert has now given some indication what he thinks needs to be done to improve the article. However, it does not seem to me that what he has said justifies the tag. If someone thinks that some topics could be included that are not yet included it seems to me there are two things to consider doing: (1) put them in or (2) suggest on the talk page that someone else do so. The fact that a couple of topics are missing does not justify a complete rewrite. As for the poor structuring, certainly the "Properties" section is an unstructured list, and could be improved by some structuring. However, I have always thought that the complete rewrite tag is for cases where the whole article is really rubbishy, not for any article that could be improved in a few respects. Does anyone else have any opinion on this? JamesBWatson (talk) 17:28, 7 December 2008 (UTC)

Yes but to tell the truth the artile should be like this since its probably as important (if not more important) than vector space.

Topology Expert (talk) 17:45, 7 December 2008 (UTC)

Yes, thank you: providing an example to show how you think the article should be structured was helpful. However, I still see the complete rewrite tag as being for cases where the whole article is really rubbishy. JamesBWatson (talk) 18:02, 7 December 2008 (UTC)

This article does not need a rewrite. Sure it's missing content, but articles needing rewrites are ones that are seriously deficient in many respects, and often have no structure at all. If you look at "vector space" before the flourish of activity that made it what it is now, it wasn't really any better than this article is now, and no rewrite was required to turn it into what it is now. This article has structure (but needs more) and content (but needs) more, that's all. And no, topological groups are not more important than vector spaces. RobHar (talk) 18:05, 7 December 2008 (UTC)

'And no, topological groups are not more important than vector spaces'. That depends on a person's opinion; I can name an important field in the theory of topological groups for every field in vector space theory you mention. In particular, I think many mathematicians will agree that Lie groups are probably the start of modern mathematics.

Topology Expert (talk) 18:57, 7 December 2008 (UTC)

Not that I really care to get in a discussion with you about this statement, but pretty much every field in math (and science) uses vector spaces. Many subjects in math use groups that are topological groups, but that doesn't mean that topological groups are that important. The amount of math that requires the notion of topological group in its full generality is extremely small compared the amount of math that uses even finite-dimensional vector spaces on a regular basis. I mean, many things in sight are topoi (including topological groups), that doesn't make them as important. Yeah, Lie groups are extremely important, and pretty cool (and I've never thought of what the "start of modern mathematics" is), but that doesn't make topological groups important. RobHar (talk) 19:10, 7 December 2008 (UTC)
Well vector spaces are completely classified in the sense that any two vector spaces over the same field, having the same dimension are isomorphic. This is not the same (there is not an analagous property) for topological groups in general.

Topology Expert (talk) 19:16, 7 December 2008 (UTC)

If you don't want the tag, remove it (as I have done). I am not in the mood to argue after the recent conflicts.

Topology Expert (talk) 19:18, 7 December 2008 (UTC)

I was simply responding to JamesBWatson's question about whether anyone else had an opinion. I gave my opinion. You didn't have to remove the tag because of that. You only argued about whether topological groups were more important than vector spaces, which has nothing to do with the tag (or really with what this talk page should be about). If you find topological groups really important then that's great for this article and you can put the energy you have into improving the article. Perhaps I shouldn't have included that last sentence in my original comment because it clearly upset you and brought the discussion even more off target, but I simply found your comment about the importance of topological groups wrong (and unnecessary). I didn't mean to upset you. Regards. RobHar (talk) 19:39, 7 December 2008 (UTC)

Template[edit]

Is Template:Basic notions in group theory intended for this article? It is referred to. Quotient group (talk) 19:27, 9 October 2009 (UTC)

Silence means consent. Quotient group (talk) 19:37, 18 October 2009 (UTC)
I'd say it usually means "apathy". Sometimes it means "consent", but often it means "you must be joking" or "I wasn't paying attention". --Zundark (talk) 20:40, 18 October 2009 (UTC)
Very often it means that the people who might have an opinion haven't seen the message. It is certainly unrealistic to assume it means consent. Since you ask I have no idea whether the template was intended for this article, nor do I care.I don't find it particularly helpful, but perhaps it does no harm. JamesBWatson (talk) 10:07, 26 October 2009 (UTC)

Non-examples?[edit]

Many examples are given of topological groups; for contrast, I would like to see non-examples. That is, are there groups that are not topological groups (aside from the discrete topology, which I guess you have for any set)? That is, is the emphasis of the definition "topological group" that there are groups without a topology, or that there is always at least one topology for a group, but that some topologies are more interesting than others? Basically, I think there is something fundamental that I don't quite grok :-) —Ben FrantzDale (talk) 14:19, 1 December 2009 (UTC)

As the article's lead says, "A topological group G is a topological space and group". A group which is not a topological space is just a group, and not a topological group. If you mean "is every group capable of having a non-trivial topology imposed on it?" then the answer is "no", as can be seen by considering a few groups of small finite order. JamesBWatson (talk) 16:13, 9 December 2009 (UTC)
Let's break the question apart. Forgive my ignorance about groups of finite order; I'm used to thinking about continuous matrix groups.
  1. Can a finite group have a topology other than the discrete topology? (As I understand it, you can have a non-trivial topology on a finite set, as defined by a graph. If that's correct, this is the same as asking if group structure on a finite set implies only a trivial topology, which I don't think is the case (as with a Rubik's cube, which is a finite group with a clear sense of what points (states) are "near" each other).)
  2. Can you give an example of one of these "few groups of small finite order"?
  3. Are there any continuous groups that don't have a non-trivial topology?
I find that an important part of understanding mathematical terms (for me) is understanding the distinction that the term is trying to make. With that in mind, it is important to understand the edge of the definition—what comes close to fitting the definition, but fails. Thanks. —Ben FrantzDale (talk) 14:19, 10 December 2009 (UTC)
  1. In the Rubik's cube clearly there is a clear sense of what points (states) are "near" each other in the same sense as that in the integers 3 and 4 are nearer to each other than 3 and 1000000, but I am not aware of any non-trivial topologisation of the Rubik cube group: can you clarify? I don't understand the relevance of the remark about graphs. Firstly, how does a graph define a topology on a finite set? Secondly, how does such a topology establish a topological group?
  2. An example of one of these "few groups of small finite order" is a cyclic group of order three. I regard both the discrete topology and the topology which makes all sets open as trivial topologies; no other topology is consistent with the group structure.
  3. I don't fully understand "Are there any continuous groups that don't have a non-trivial topology?" The only sense I can make of it is "are there any continuous groups that do have a trivial topology", in which case any group can clearly be made into a topological group by giving it the topology in which all sets are open. JamesBWatson (talk) 15:31, 14 December 2009 (UTC)
Ad 1: topologies on a finite group G that make it a topological group are in 1-to-1 correspondence with normal subgroups of G. On the one hand, if H is a normal subgroup of G, it defines a topological group on G whose open sets are unions of cosets of H. On the other hand, every topological group on G is of this form (H is the smallest neighbourhood of 1). In general, topologies on a finite set correspond to preorders, but in the case of topological groups, the preorder is always an equivalence relation (in fact, a congruence relation on the group). — Emil J. 17:14, 14 December 2009 (UTC)
I guess one way to look at this is to consider the question does every group admit a topology compatible with the group structure? - the answer is trivially yes because of the discrete topology. Then the question becomes not one of "finding non-examples" but one of enumerating and characterising the possible topological group structures having the same underlying group. EmilJ gives an example of such a characterisation (in terms of normal subgroups) for the case of finite groups. Miguel (talk) 22:59, 14 December 2009 (UTC)