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Thread: Open sets

  1. #1
    Aug 2007

    Open sets

    Theorem: Every open set  U \subset \mathbb{R} can be uniquely expressed as a countable union of disjoint open intervals. The endpoints of  U do not belong to  U (Pugh, p. 63).

    The book goes on to prove this by first defining  a_x = \inf\{ a :  \exists(a,x) \subset U \} and  b_x = \sup \{b: \exists (x,b) \subset U \} and then an interval  I_x = (a_x, b_x) .

    I don't understand why they define  \sup and  \inf for those particular sets. What does  I_x represent? I know that its the largest interval, because the endpoints are the glb and lub.
    Last edited by shilz222; Sep 10th 2007 at 01:56 PM.
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  2. #2
    MHF Contributor

    Aug 2006
    Once again, I am not quite sure about your misunderstanding.
    What the text is doing is constructing what are know as components in topology. A component is a maximal connected set. Components are pair-wise disjoint. In \Re^1 the components of an open subspace are open intervals. You are correct to worry about glbís and lubís. In the example you gave, unless U is bounded, it is possible that a_x  =  - \infty \mbox{  or  } b_x  = \infty .
    Does this help you?
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