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experimentX Group Title

Q1: An open interval in \( \mathbb R^1 \) is not open in \( \mathbb R^2 \), then what is it? Q2: In Cantor intersection theorem, if each Q is were non empty open set then would the intersection be empty or non-empty?

  • one year ago
  • one year ago

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  1. Jemurray3 Group Title
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    You're asking how to classify a line segment that doesn't contain its endpoints if it's embedded in R2?

    • one year ago
  2. experimentX Group Title
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    kinda yes!! http://en.wikipedia.org/wiki/Open_set#Open_and_closed_are_not_mutually_exclusive

    • one year ago
  3. Jemurray3 Group Title
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    Are you okay with why it's definitely not open?

    • one year ago
  4. experimentX Group Title
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    it does not contain it's end point in R^1, but it's an open set in R^1, my book says the the interval is no longer open set in R^2 because it cannot contain open two ball in R^2 space inside it.

    • one year ago
  5. experimentX Group Title
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    i am guessing it to be neither open nor closed set.

    • one year ago
  6. Jemurray3 Group Title
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    I would agree. For a set to be open, there needs to exist a tiny little interval around every point in the set such that all points in the interval also belong to the set. in R1, that looks like this: |dw:1356810565391:dw|

    • one year ago
  7. Jemurray3 Group Title
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    But in R2, you need a circle, not just an interval: |dw:1356810632181:dw|

    • one year ago
  8. Jemurray3 Group Title
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    In R3 you need a sphere, etc etc etc. And the definition of a closed set is a set such that its complement is an open set, which is also false in this case because the endpoints of the interval are contained in the complement of the set, so it contains some limit points.

    • one year ago
  9. Jemurray3 Group Title
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    Circle and Sphere are actually misleading. I should have said Disk and Ball.

    • one year ago
  10. experimentX Group Title
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    According to the definition on Wikipedia, this can't be clopen either.

    • one year ago
  11. Jemurray3 Group Title
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    That's right. A clopen set is both open and closed, this set is neither open nor closed.

    • one year ago
  12. experimentX Group Title
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    Let's see what other people have to say ... i'll leave the post open

    • one year ago
  13. experimentX Group Title
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    Any idea on Cantor intersection theorem?

    • one year ago
  14. Jemurray3 Group Title
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    What about it?

    • one year ago
  15. experimentX Group Title
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    what if we assume that each Qk were open and not empty? what would be the final result of the intersection of Qk's?

    • one year ago
  16. Jemurray3 Group Title
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    Then it may well be empty. Are you familiar with the proof of this theorem?

    • one year ago
  17. experimentX Group Title
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    yes,, it creates a sequence from each Qk's .. and as k tends to increase to inf ,,, by bolzano Weirstrass this would be accumulation point. Since, each Qk is closed and closed set contain at least one accumulation points, at least there is one element in the intersection.

    • one year ago
  18. Jemurray3 Group Title
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    Right. If the sets are open, though, it's possible that the sequence of infima converges to an accumulation point that doesn't actually belong to the sets.

    • one year ago
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