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Open Sets Span

  1. Sep 6, 2008 #1
    Am I correct in thinking that the union of disjoint open sets cannot span a continuous interval? Assume that each of the sets is a proper subset of the interval. Does this apply even if the collection of open sets is uncountable infinite?
     
  2. jcsd
  3. Sep 7, 2008 #2

    CompuChip

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    You are correct in thinking that, as long as the interval is closed of course. In fact, you cannot even do it with non-disjoint sets. Otherwise, it would mean that the set [0, 1] is open in R (see the properties of a topology).

    For an open interval, it is trivial, e.g. ]0, 1[ is spanned by a single disjoint open set.
     
  4. Sep 7, 2008 #3

    CRGreathouse

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    What about this: Can a single open interval be spanned by two or more nontrivial disjoint open subsets of the interval?
     
  5. Sep 7, 2008 #4

    CompuChip

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    CRGreathouse, the answer seems to be quite trivially "no". So probably I am missing here, and my guess the problem is in the word "span".

    What exactly is meant by "spanning" in this context?
     
  6. Sep 7, 2008 #5

    gel

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    I wouldn't call it trivial.
    Still, not too difficult - note that the interval between any two disjoint open intervals is a closed interval. So the problem reduces to the case of a closed bounded interval, which is compact.
     
  7. Sep 7, 2008 #6

    CRGreathouse

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    First of all, I have no interest in the answer -- I just thought this may have been the question intended (though not written!) by the OP.

    But I simply meant for the union of the subsets to be the full set. I agree that this appears trivially impossible.
     
  8. Sep 7, 2008 #7

    morphism

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    An open interval can't be a nontrivial disjoint union of open sets because it's connected.
     
  9. Sep 7, 2008 #8

    gel

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    of course, that's the simple answer:)
     
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