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Thread: analysis question

  1. #1
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    analysis question

    Show that every open set UR is a union of at most countably many disjoint open intervals [Hint: Each point x of U lies in a unique maximal interval of U, which is the union of all intervals contained in U which contain x. Each such interval contains at least one rational number.]

    can anyone help me to approach this question ? ,,,
    thanks a lot
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  2. #2
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    Quote Originally Posted by jin_nzzang View Post
    Show that every open set UR is a union of at most countably many disjoint open intervals [Hint: Each point x of U lies in a unique maximal interval of U, which is the union of all intervals contained in U which contain x. Each such interval contains at least one rational number.]
    This problem is completely solved by using the hint.
    The interesting part of this is proving the hint is true.
    If $\displaystyle x \in U$ define two sets $\displaystyle L_x = \left\{ {y:\left[ {x,y} \right) \subset U} \right\}\quad \& \quad K_x = \left\{ {y:\left( {y,x} \right] \subset U} \right\}$.
    You be able to give reasons why both sets are nonempty.

    Now look for least upper bound of $\displaystyle L_x$ and greatest lower bound of $\displaystyle K_x$ and with those as endpoints you have a maximal open interval containing $\displaystyle x$ and is a subset of $\displaystyle U$.
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