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Open sets

  1. Feb 13, 2008 #1
    1. The problem statement, all variables and given/known data
    If you have a collection of disjoint open sets in a general topological space whose union is open, is it true that each of them individually must be open? Why?

    EDIT: this makes absolutely no sense. here is what I meant to ask:
    EDIT:If you have a collection of disjoint sets in a general topological space whose union is open, is it true that
    EDIT:each of them individually must be open? Why?

    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Feb 13, 2008
  2. jcsd
  3. Feb 13, 2008 #2
    As stated the answer is trivially yes because they are disjoint and OPEN. Did you really mean to ask if you had a collection of disjoint set's whose union in a topological space is open is each necessarily open. In that case the answer is false consider the trivial topology on {a,b,c}. Then the collection {{a}, {b}, {c}} is a collection of disjoint sets whose union is open but none of the ndividual sets is open.
     
  4. Feb 13, 2008 #3
    What if we are in a locally path connected and path connected space?
     
  5. Feb 13, 2008 #4
    What are the original sets that comprise the union? They can't be open individually if you are being asked that very question.
     
  6. Feb 13, 2008 #5
    Sorry. Reread the EDIT. I am foolish.
     
  7. Feb 13, 2008 #6

    NateTG

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    Consider any non-open set [itex]X[/itex] and it's complement [itex]\bar{X}[/itex].
    Their union is clearly open since it's the entire space, they're disjoint by construction, and [itex]X[/itex] is non-open by hypothesis.
     
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