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Closed Set

  1. Nov 4, 2015 #1
    Hi,
    How and why set of natural numbers is closed?
     
  2. jcsd
  3. Nov 4, 2015 #2

    micromass

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    Closed in what topological/metric space?

    What are your thoughts?
     
  4. Nov 4, 2015 #3
    Good question. I think it's about metric space.
     
  5. Nov 4, 2015 #4

    micromass

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    What metric space?
     
  6. Nov 4, 2015 #5

    WWGD

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    OP:There is no natural definition of natural numbers in a generic metric space. Do you have any particular embedding of the natural numbers in mind?
     
  7. Nov 4, 2015 #6

    fresh_42

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    ℕ as a half-group is closed under addition.
    ℕ as a discreet topological space is closed by itself. A discreet metric won't change anything. But both is more or less trivial.
    ℕ⊆ℝ is closed since its complement is open, i.e. you can find to each real number r, that is not natural, an open intervall that contains r but still no natural number.
     
  8. Nov 4, 2015 #7

    HallsofIvy

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    Every topological space is "closed' as a subset of itself. If you have it embedded in the real numbers with the "usual metric", d(x, y)= |x- y|, then it is closed as fresh_42 says.
     
  9. Nov 4, 2015 #8

    WWGD

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    I assume you are replying to my post. There is no assumed embedding of the naturals into the generic metric space.
     
  10. Nov 4, 2015 #9

    fresh_42

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    Yes, you are right. And I can imagine a couple of very funny embeddings, metric or not. But considering the simplicity of the question it's not very unlikely that ℕ⊂ℝ with it's euclidean metric is meant. And yes, it hasn't been mentioned. Reading the questions here I found that most of them are far from being precise or even clear.
     
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