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Math Help - Don't understand the proof of a simple theorem

  1. #1
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    Don't understand the proof of a simple theorem

    I found the theorem and its proof in the attachment file.

    The thing is that I don't understand why suddenly "either disjoint or equal" become "disjoint" for "Ix=Iy" condition.

    Could you explain this for me?
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  2. #2
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    Quote Originally Posted by gyro73 View Post
    I found the theorem and its proof in the attachment file. The thing is that I don't understand why suddenly "either disjoint or equal" become "disjoint" for "Ix=Iy" condition.
    It is actually easy. But you need to understand that I_x is a connect set.
    In R the open connected sets are (-\infty ,a),~(b, \infty ),\text{ or }(a,b).
    If two connected set have a point in common there union is connected and has one of those three forms.
    Now by definition of I_x is a maximum connected open set.
    Thus, either I_x=I_y\text{ or }I_x\cap I_y=\emptyset.
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