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Clopen Set

  1. Sep 30, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that if A ⊆ ℝ is both open and closed then A is either ℝ or ∅.

    2. Relevant equations

    G ∩ bd(G) = ∅ ⇒ G is open

    bd(F) ⊆ F ⇒ F is closed

    bd(S) = bd(ℝ∖S) = bd(S')

    3. The attempt at a solution

    Suppose A is a clopen set such that it is neither ℝ nor ∅ then ℝ∖A = A' is neither ℝ nor ∅. Now, A' is open because A is open (and closed). So ℝ = A ∪ A', where both A and A' are closed, which implies that ℝ is closed, a contradiction.
     
  2. jcsd
  3. Sep 30, 2011 #2

    SammyS

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    Why is the fact that "ℝ is closed" a contradiction. ℝ is indeed closed.
     
  4. Sep 30, 2011 #3
    Isn't ℝ a clopen set?
     
  5. Sep 30, 2011 #4

    SammyS

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    What's the definition of a "clopen" set?
     
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