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

by marvel_boy 11y ago

ihm 11y ago
It's locally R and connected but not path connected.
- jordigh 11y ago
  Huh, why not? Because paths must have the real numbers as their domain?
  - HerrMonnezza 11y ago
    Yes, exactly. Paths are defined as continuous maps from the closed real interval [0,1] to a space.
jordigh 11y 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 11y 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 11y ago
  Euclidean plane? Not line? Did I misunderstand the construction?
  - contravariant 11y ago
    No, he probably meant the euclidean line.
ggchappell 11y 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.)

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

ihm 11y ago

It's locally R and connected but not path connected.

jordigh 11y ago
Huh, why not? Because paths must have the real numbers as their domain?
- HerrMonnezza 11y ago
  Yes, exactly. Paths are defined as continuous maps from the closed real interval [0,1] to a space.

jordigh 11y 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 11y 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 11y ago
Euclidean plane? Not line? Did I misunderstand the construction?
- contravariant 11y ago
  No, he probably meant the euclidean line.

ggchappell 11y 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.)