Added details for the proof that the long line is not contractible.

]]>Interesting question. It does seem like it ought to be a “long ray” that’s also “locally long” in that it’s isomorphic to its own subintervals $[n,n+1)$.

]]>Musing about my own question, maybe the answer is located in some subfield of the surreal numbers…

]]>The usual line (or rather, ray) $[0, \infty)$, as an ordered set, has a nice coalgebraic description: it is the terminal coalgebra for the functor $F: Pos \to Pos$ that sends a poset $X$ to $\mathbb{N} \times X$ endowed with the lexicographic order. (See continued fraction.) Does anyone have an idea what the terminal coalgebra for $X \mapsto \omega_1 \times X$ (the latter with lexicographic order) looks like? I don’t suppose that it’s the long ray. Something like a super-long ray??

]]>Thanks. I guess that means that you thought you had created that page sometime in the past, but hadn’t. I was just puzzled by the idea that while there must be hundreds of places scattered around the nLab where one refers to the product of spaces, somehow long line created a sudden urge to refer to the “Tychonoff product”! :-) But now I see that’s probably not what happened here.

]]>It’s only need to link to the correct page. But that page is not there! I will make it.

]]>While fixing a mistake at long line, I noticed that at the misstated property, Toby had edited to “Tychonoff product”. Is that supposed to be the same as the categorical product for $Top$? If so, I don’t see how the ’Tychonoff’ is really needed.

]]>Thanks, Mike. Perhaps one doesn’t see many references because the interest in the long line is that it’s a topological manifold, but the ’long circle’ (under whatever definition) isn’t.

]]>There’s a question of whether the long circle should be the one point compactification of the long line, or the result of gluing the long ray into a loop. Note that some people like Steen and Seebach use ’long line’ to mean the long ray. I don’t know a reference for ’long circle’, but its use to mean something like this seems fairly obvious.

]]>I’ve added to the formerly stubby long line.

Incidentally, I thought the one-point compactification of the long line was called the “long circle”, but I don’t see mention of that via google. What’s that thing called?

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