Talk:Long line (topology)

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Contents 1 Opinion 2 Question 3 Characterization of 1dim. manifolds? 4 Short Ray

[edit] 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)

[edit] Question

I do not really understand why if you try to make a long line using more than ω₁ intervals you get something that is no more locally homeomorphic to . Why is that? Cthulhu.mythos 08:49, 1 June 2006 (UTC)

Well one way to see it is that the topolgy on 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(ω₁) = ω₁. Cool! Cthulhu.mythos 15:45, 1 June 2006 (UTC)

That's not right. cof(ω₁) = 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)

[edit] 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)

[edit] 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))

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