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Math Help - Sets

  1. #1
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    Sets

    "Every open subset O of R is an at most countable union of pairwise disjoint open intervals (a,b). "

    Could someone explain why this is true? If every open intervals are disjoint, e.g. (1,2) and (2,3), does that mean the point in between, in this case 2, is not included?

    Thank you.

    Felix
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  2. #2
    Member
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    Quote Originally Posted by felixmcgrady View Post
    "Every open subset O of R is an at most countable union of pairwise disjoint open intervals (a,b). "

    Could someone explain why this is true? If every open intervals are disjoint, e.g. (1,2) and (2,3), does that mean the point in between, in this case 2, is not included?

    Thank you.

    Felix
    Given an open subset O of R and any two elements a, b of O, define a ~ b iff there is an open interval in O containing a and b. It's easy to show that ~ is an equivalence relation, and that the equivalence classes it induces are open and connected. Hence we can decompose O into disjoint connected open components, of which there can be only countably many. (If there were uncountably many, there'd have to be some number L such that uncountably many of the components had length at least L. But you can't squeeze uncountably many line segments of length L into the real line without some overlap.)
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