Talk:Long line (topology)

History

Does anyone know more about the history of this counterexample? It would be an interesting addition to the article. —Preceding unsigned comment added by 24.84.204.127 (talk) 18:11, 2 April 2011 (UTC)

Original Research related to this subject

I have discovered a different and MORE ELEGANT class of LONG LINES. I am looking for somebody, who would co-author a paper about it with me, by taking the ideas from me and writing the paper, while putting my name as the name of the first author. H. Tomasz Grzybowski tel. +48-780-129-544 e-mail: htg@interia.pl

Opinion

Good article! Can we put it on the main page? (haha) 24.7.87.135 09:54, 6 March 2006 (UTC)

Uh? How sarcastic is that comment supposed to be? If you don't like this article, please improve it, or at least state how you think it should be improved. --Gro-Tsen 13:07, 6 March 2006 (UTC)

I was just saying that it's worthy of being put on the front page but we couldn't actually do it. So I was congratulating those who have worked on it. 24.7.87.135 04:30, 13 March 2006 (UTC)

It's not main page worthy yet, but maybe one day. --C S (talk) 02:22, 12 May 2008 (UTC)

Question

I do not really understand why if you try to make a long line using more than $\omega _{1}$ intervals you get something that is no more locally homeomorphic to $\mathbb {R}$. Why is that? Cthulhu.mythos 08:49, 1 June 2006 (UTC)

Well one way to see it is that the topolgy on $\mathbb {R}$ has a countable base, while the topology on the long line does not, so they cannot be homeomorphic. -lethe ^{talk +} 10:49, 1 June 2006 (UTC)

Ahh, I read the sentence in the article, and I think I didn't answer your question. Let me try again. So [0,1)×ω₁ is locally homeomorphic to R but not homeomorphic, which I answered above, but perhaps you saw already. Now you want to know why [0,1)×(ω₁+1) (or any other ordinal larger than ω₁) is not even locally homeomorphic to R, is that it? Well, the ordered pair (0,ω₁) is an element of [0,1)×(ω₁+1), and this point does not even have a countable local base, so this space cannot be even locally homeomorphic to R. This point sits atop an uncountable sequence of lower ordinals, something that happens in the neighborhood of no point in R. -lethe ^{talk +} 10:57, 1 June 2006 (UTC)

Yeppp. I see, that's because $cof(\omega _{1})=\omega _{1}$. Cool! Cthulhu.mythos 15:45, 1 June 2006 (UTC)

That's not right. $cof(\omega _{1})$ = continuum. Cofinality is a cardinal number. Leocat 01:09, 18 October 2006 (UTC)

That's exactly right. And ω₁ is an element of any larger ordinal. The space ω₁ (with the order topology) is first-countable but not second-countable, while any larger ordinal is neither first- nor second-countable. The relevant statements about long lines follow. -lethe ^{talk +} 18:06, 1 June 2006 (UTC)

If one considers intervals (0,1], instead of [0,1), the pathology of local uncountable basis disappear and you can use any ordinal to build very long lines, why don't we put that in the article? —Preceding unsigned comment added by 137.204.135.219 (talk) 14:36, 22 July 2009 (UTC)

Because it's not really a connected line since it doesn't have the least upper bound property —Preceding unsigned comment added by Standard Oil (talk • contribs) 05:48, 25 July 2009 (UTC)

Characterization of 1dim. manifolds?

I may be mistaken but I think the circle is missing in the line of the article "It makes sense to consider all the long spaces at once, however, because every connected (non-empty) one-dimensional (not necessarily separable) topological manifold possibly with boundary, is homeomorphic to either the closed interval, the open interval (real line), the half-open interval, the closed long ray, the open long ray, or the long line." --84.167.88.110 18:08, 29 July 2006 (UTC)

You're right. It's fixed now. AxelBoldt (talk) 17:51, 2 February 2009 (UTC)

Short Ray

So ω₀ × [0, 1) is just a simple closed ray, right? (Sorry, I don't really know this stuff, I just heard about the possible proof of the Poincaré conjecture and started reading about it --80.175.250.218 12:32, 23 August 2006 (UTC))

$\omega \times [0,1)$ is identical (homeomorphic) to $[0,1)$ itself. Yes, you might call it the (closed) short ray. --Gro-Tsen 17:31, 23 August 2006 (UTC)

In fact, $\alpha \times [0,1)$ is order-isomorphic to $[0,1)$ for any countable ordinal $\alpha$. Isn't this already in the article? --Sniffnoy 18:37, 23 August 2006 (UTC)

Riemannian metric?

Can we equip the long line with a Riemannian metric? It would be nice to add this information to the article. AxelBoldt (talk) 00:15, 6 October 2008 (UTC)

Unless I am mistaken, giving the long line a Riemannian metric (i.e., a nonvanishing section of the tensor square of its tangent/cotangent bundle) is basically the same as finding a nonvanishing section of its tangent/cotangent bundle (think 1-dimensional!); and I believe that can't be done, because by integrating, say, a nonvanishing vector field, and passing to the limit we'd find a point where it has to vanish. If I find the time, I'll try to check this carefully and update the article (but if someone is more sure than I am, I'll gladly let him do it). --Gro-Tsen (talk) 14:52, 6 October 2008 (UTC)

You are right, it's not possible. I found an argument in James A. Morrow, "Shorter Notes: The Tangent Bundle of the Long Line", Proceedings of the American Mathematical Society, Vol. 23, No. 2 (Nov., 1969), p. 458 [1]: Riemann manifolds, even if not assumed to be paracompact, can be shown to be metric spaces, but the long line is not metrizable. AxelBoldt (talk) 00:06, 7 October 2008 (UTC)

Differentiable structures

What does the article mean when it says that the long line has several differentiable structures? Let L be the long line with one differentiable structure and L' with another. That L and L' are different can mean either that there is no C^r-diffeomorphism L -> L' (strong sense) or that the identity map L -> L' is not a C^r-diffeomorphism (weak sense). I remember that the long ray has several different real analytic structures in the strong sense, but how about C^r structures? And what about the long line? Anyway it would be nice if the article discussed also this thing.128.214.51.113 (talk) 11:25, 3 April 2009 (UTC)

I should hope it means the strong sense, because in the weak sense you define even the real line has several differentiable structures (take any homeomorphism of the real line which is not a diffeomorphism and use it to transport the usual differentiable structure: this gives another differentiable structure, which is diffeomorphic to the usual one, but such that the identity map is not a diffeomorphism). Certainly the existence of different differentiable structures on the long ray/line must mean something more interesting than that. But I no longer have the proof in mind, so I guess someone should re-read the Koch & Puppe article and check what they do exactly and how they do it. --Gro-Tsen (talk) 10:22, 5 April 2009 (UTC)

Path-Connectedness

I came by this page a while back and corrected the statement that the long line is path-connected. (I deleted it.) The long line is very clearly not path-connected. The assertion that it is path-connected has reappeared. It would be helpful if people didn't resurrect errors after they have been corrected. —Preceding unsigned comment added by 173.253.136.180 (talk) 23:05, 10 April 2011 (UTC)

You are wrong, the Long Line is path-connected (provided you don't extend it by adding a point "at infinity"); this follows from the fact that for any countable ordinal α, the cartesian product α×[0;1) is homeomorphic to [0;1) (so that you can connect 0 to an arbitrary point on the long line). And Paul August, who reverted your change, pointed out in his comment to the revert that the fact is stated in the classic book by Steen & Seebach already in the references of the article. --Gro-Tsen (talk) 00:46, 11 April 2011 (UTC)

Withdrawn. I can see that there's a flaw in my argument now. However, a comment illustrated by this incident that applies to this entire article: specific references and page numbers should be attached to specific claims. Otherwise, you're just making a lot of claims and sending the curious reader to a stack of books for verification. If this claim had been properly attached to a reference (preferably something it is possible to find in print)---or, you know, proved---I would have read the proof. (But I'm not interested in a debate, and I won't be back.) —Preceding unsigned comment added by 173.253.211.226 (talk) 14:30, 11 April 2011 (UTC)

External Link

I'd like to add a link to The Long Line on the Brubeck Topology Database, but I wanted to make sure that it was appropriate to do so first. It contains fully traceable proofs of why the Long Line does or doesn't satisfy ~60 different topological properties, so I believe it is "neutral and accurate material that is relevant to an encyclopedic understanding of the subject and cannot be integrated into the Wikipedia article due to ... amount of detail" and thus meets the guidelines for inclusion. Any objections? Jamesdabbs (talk) 22:02, 17 June 2012 (UTC)

Be bold! --Gro-Tsen (talk) 09:45, 6 July 2012 (UTC)

the database is offline Mennucc (talk) 09:22, 10 January 2014 (UTC)

Different Smooth Structures

In the currect version of the article, it is not clear what is meant that there are different smooth structures. It is stated that every C^k-structure gives rise to infinitely many C^{k+1} structures.

• If just means "different", then this is a boring result since this is also true for the real line
• If this means "pairwise non diffeomorphic", then I am not sure it is true for k>=1.

Therefore I changed it to infinitely many (in fact 2^aleph1) smooth (C^infty) structures giving the same topology (C^0). This is true (I have a reference for it), surprising (it is not true for the real line). I also changed the wording at the similar part where we talk about analyticity.

The stronger version: Every C^k-structure gives rise to pairwise non-diffeomorphic C^{k+1} structures... I think I read somewhere that this is not true. That for a Hausdorff-manifold (without requiring paracompactness or second countability) the C^1-structure always determines the C^infty-strucutre up to diffeomorphism. But I don't remember where I read that.

That's why I changed the article the way I did. --129.13.187.218 (talk) 15:01, 13 January 2020 (UTC)

Stone-Čech compactification

The page says: "The extended long ray L ∗ {\displaystyle L^{*}} L^{*} is compact. It is the one-point compactification of the closed long ray L , {\displaystyle L,} L, but it is --No !-- also its Stone-Čech compactification, because any continuous function from the (closed or open) long ray to the real line is eventually constant."

I agree that for the closed long ray, the Alexander and Čech-Stone compactifications agree, for the reason given. However, for the long line, the two limits do not have to agree, so Čech-Stone compactification adds two points. 178.255.168.156 (talk) 16:23, 20 November 2021 (UTC)

Assertion on continuous functions

For the assertion:

"In fact, every continuous function is eventually constant."

While I obviously agree with the earlier statement that all increasing functions are eventually constant, any continuous non-constant function periodic over a unit interval (e.g. sin(x/2pi)) is both well-defined in this domain and never constant. Is this a mistake?

Or is this just saying the function can't truly be continuous at limit points of the construction, which still seems misleading? TricksterWolf (talk) 01:55, 4 February 2024 (UTC)