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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?
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.
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.
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.
It’s only need to link to the correct page. But that page is not there! I will make it.
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.
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??
Musing about my own question, maybe the answer is located in some subfield of the surreal numbers…
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 $[n,n+1)$.
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