4 ms·

Long line (topology)

46 points by luu 12y ago

airza 12y ago
it's really weird to see a euclidian-looking line that has no strictly increasing sequences or functions..
transfire 12y ago
Closely associated with the Navel Line of Sight.
marvel_boy 12y ago
Newbie here. What is the counterexample of Topolopy Long Line serves?
- ihm 12y ago
  It's locally R and connected but not path connected.
  - jordigh 12y ago
    Huh, why not? Because paths must have the real numbers as their domain?
    - HerrMonnezza 12y ago
      Yes, exactly. Paths are defined as continuous maps from the closed real interval [0,1] to a space.
- jordigh 12y ago
  It is very much not compact. In fact, It is not Lindelöf, a weaker form of compactness: https://en.wikipedia.org/wiki/Lindel%C3%B6f_space https://en.wikipedia.org/wiki/Lindel%C3%B6f_space It is also not separable, i.e. no countable everywhere dense subset. I've added this to the Wikipedia page.
- fantispug 12y ago
  It is a Hausdorff space that is locally homeomorphic to the Euclidean plane, but not paracompact. Manifolds are usually defined to be paracompact because this implies the existence of partitions of unity which have many nice implications: http://math.stackexchange.com/questions/98105/why-are-smooth-manifolds-defined-to-be-paracompact http://math.stackexchange.com/questions/98105/why-are-smooth...
  - jordigh 12y ago
    Euclidean plane? Not line? Did I misunderstand the construction?
    - contravariant 12y ago
      No, he probably meant the euclidean line.
- ggchappell 12y ago
  An ELI5 answer: Think about "talking a walk" along the Euclidean Line -- the real-number line, if you like. Face in one direction, and then start taking steps. Stepping Example #1. Start at 0 and face toward the positive numbers. Step to 1. Step to 2. Step to 3. Step to 4. Etc. Each time you take a step, just a bit more of the line is behind you. And if I pick a point on the line, then no matter which point I pick, this way of stepping will eventually put that point behind you. Stepping Example #2. Start at 0 and face toward the positive numbers. Step to 1/2. Step to 2/3. Step to 3/4. Step to 4/5. Etc. Again, each time you take a step, just a bit more of the line is behind you. But this time, I can find points -- 6, for example, or 300 million -- that will never be behind you. (There is a technical term for the first kind of stepping. We say that the set of points you step on, is cofinal in the order on the Euclidean Line. The second kind of stepping does not have this property.) One of the curious properties of the Long Line is, first, that "locally", it is exactly like the Euclidean Line. Stand anywhere on the Long Line and look around. "Looks like I'm on the Euclidean Line," you say. But, second, every possible way of stepping ends up like Example #2 above. No matter what steps you choose to take, I can find a point -- lots of points, in fact -- that will never be behind you. And that is why we say it is "long". (Formally, the Long Line is locally homeomorphic to the Euclidean Line, but the order on the Long Line admits no countable cofinal subset.)
kazinator 12y ago
Insert "come from a long line of topologists" jokes here.
olivermarks 12y ago
I am baffled by this and feeling stupid
- jordigh 12y ago
  There's really no reason to feel stupid just because you don't know what the terminology means, if that is indeed the problem. Would you feel stupid if you were to walk into a Scientology meeting and you didn't know what SP, Xenu, or fair game meant? I wouldn't. It's just a bunch of jargon that I would have to pick up if I were really interested in knowing the underlying concepts being discussed (and I'm not, which makes me feel even less stupid for not wanting to learn this). And even if you know what all the words mean, it can take some time and effort to put them together, even for practitioners.
  - westoncb 12y ago
    The situation is even trickier than that because you can keep on looking up these terms, one after another, but it's going to take a lot more than their particular definitions before things begin 'coming together' for someone. I think this largely comes down to the fact that there's no place people can go to learn what it is mathematics is, what it's about, how it works, etc.—and consequently, what mathematicians are interested in and why they have their particular quirks of formulation, and why certain points (which can seem utterly uninteresting to an outsider) are emphasized. Of course you can get this by doing a math degree (and maybe graduate work), but I'm just trying to describe the difficulty in teaching oneself. At least in the U.S., our exposure to mathematics in formal education (pre-university), gives exactly all of the wrong answers to those questions above, so that when you try to start teaching yourself you never know what direction to search in in order to resolve ambiguities.
- gjulianm 12y ago
  I wouldn't worry about that. I'm studying mathematics and I also feel stupid while reading new concepts on Wikipedia. The problem with mathematics is that, if you don't know the idea behind the concept or which kind of problem is it trying to solve, it's almost impossible to really understand the definitions. And even when you understand the concepts and ideas, it can still be difficult to understand the definitions. Few months ago, a professor told us the 100% formal definition of a function: it wasn't easy to understand and we all had this "wtf" face.

46 points by luu 12y ago

airza 12y ago

it's really weird to see a euclidian-looking line that has no strictly increasing sequences or functions..

transfire 12y ago

Closely associated with the Navel Line of Sight.

marvel_boy 12y ago

Newbie here. What is the counterexample of Topolopy Long Line serves?

ihm 12y ago
It's locally R and connected but not path connected.
- jordigh 12y ago
  Huh, why not? Because paths must have the real numbers as their domain?
  - HerrMonnezza 12y ago
    Yes, exactly. Paths are defined as continuous maps from the closed real interval [0,1] to a space.
jordigh 12y ago
It is very much not compact. In fact, It is not Lindelöf, a weaker form of compactness: https://en.wikipedia.org/wiki/Lindel%C3%B6f_space https://en.wikipedia.org/wiki/Lindel%C3%B6f_space It is also not separable, i.e. no countable everywhere dense subset. I've added this to the Wikipedia page.
fantispug 12y ago
It is a Hausdorff space that is locally homeomorphic to the Euclidean plane, but not paracompact. Manifolds are usually defined to be paracompact because this implies the existence of partitions of unity which have many nice implications: http://math.stackexchange.com/questions/98105/why-are-smooth-manifolds-defined-to-be-paracompact http://math.stackexchange.com/questions/98105/why-are-smooth...
- jordigh 12y ago
  Euclidean plane? Not line? Did I misunderstand the construction?
  - contravariant 12y ago
    No, he probably meant the euclidean line.
ggchappell 12y ago
An ELI5 answer: Think about "talking a walk" along the Euclidean Line -- the real-number line, if you like. Face in one direction, and then start taking steps. Stepping Example #1. Start at 0 and face toward the positive numbers. Step to 1. Step to 2. Step to 3. Step to 4. Etc. Each time you take a step, just a bit more of the line is behind you. And if I pick a point on the line, then no matter which point I pick, this way of stepping will eventually put that point behind you. Stepping Example #2. Start at 0 and face toward the positive numbers. Step to 1/2. Step to 2/3. Step to 3/4. Step to 4/5. Etc. Again, each time you take a step, just a bit more of the line is behind you. But this time, I can find points -- 6, for example, or 300 million -- that will never be behind you. (There is a technical term for the first kind of stepping. We say that the set of points you step on, is cofinal in the order on the Euclidean Line. The second kind of stepping does not have this property.) One of the curious properties of the Long Line is, first, that "locally", it is exactly like the Euclidean Line. Stand anywhere on the Long Line and look around. "Looks like I'm on the Euclidean Line," you say. But, second, every possible way of stepping ends up like Example #2 above. No matter what steps you choose to take, I can find a point -- lots of points, in fact -- that will never be behind you. And that is why we say it is "long". (Formally, the Long Line is locally homeomorphic to the Euclidean Line, but the order on the Long Line admits no countable cofinal subset.)

kazinator 12y ago

Insert "come from a long line of topologists" jokes here.

olivermarks 12y ago

I am baffled by this and feeling stupid

jordigh 12y ago
There's really no reason to feel stupid just because you don't know what the terminology means, if that is indeed the problem. Would you feel stupid if you were to walk into a Scientology meeting and you didn't know what SP, Xenu, or fair game meant? I wouldn't. It's just a bunch of jargon that I would have to pick up if I were really interested in knowing the underlying concepts being discussed (and I'm not, which makes me feel even less stupid for not wanting to learn this). And even if you know what all the words mean, it can take some time and effort to put them together, even for practitioners.
- westoncb 12y ago
  The situation is even trickier than that because you can keep on looking up these terms, one after another, but it's going to take a lot more than their particular definitions before things begin 'coming together' for someone. I think this largely comes down to the fact that there's no place people can go to learn what it is mathematics is, what it's about, how it works, etc.—and consequently, what mathematicians are interested in and why they have their particular quirks of formulation, and why certain points (which can seem utterly uninteresting to an outsider) are emphasized. Of course you can get this by doing a math degree (and maybe graduate work), but I'm just trying to describe the difficulty in teaching oneself. At least in the U.S., our exposure to mathematics in formal education (pre-university), gives exactly all of the wrong answers to those questions above, so that when you try to start teaching yourself you never know what direction to search in in order to resolve ambiguities.
gjulianm 12y ago
I wouldn't worry about that. I'm studying mathematics and I also feel stupid while reading new concepts on Wikipedia. The problem with mathematics is that, if you don't know the idea behind the concept or which kind of problem is it trying to solve, it's almost impossible to really understand the definitions. And even when you understand the concepts and ideas, it can still be difficult to understand the definitions. Few months ago, a professor told us the 100% formal definition of a function: it wasn't easy to understand and we all had this "wtf" face.