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Math Help - a strange question

  1. #1
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    a strange question

    If E is a connected subset of M, and if A and B are disjoint open sets in M with E \subsetAUB, prove that either E \subsetA or E \subset B.

    since we already know E \subsetAUB, and A and B are disjoint, so E must be a subset of A or B, why do we even need the connectedness condition here?
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  2. #2
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    Re: a strange question

    Quote Originally Posted by wopashui View Post
    If E is a connected subset of M, and if A and B are disjoint open sets in M with E \subsetAUB, prove that either E \subsetA or E \subset B.
    since we already know E \subsetAUB, and A and B are disjoint, so E must be a subset of A or B, why do we even need the connectedness condition here?
    Let M=[-4,4],~E=[-2,-1]\cup[1,2], then if A=(-3,0)~\&~B=(0,3) then E\subset A\cup B.

    That is why we require E to be connected.
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    Re: a strange question

    Quote Originally Posted by Plato View Post
    Let M=[-4,4],~E=[-2,-1]\cup[1,2], then if A=(-3,0)~\&~B=(0,3) then E\subset A\cup B.

    That is why we require E to be connected.
    hmm ic, so how can I use the connectedness to prove this?
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  4. #4
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    Re: a strange question

    Quote Originally Posted by wopashui View Post
    so how can I use the connectedness to prove this?
    Let E_A=E\cap A~\&~E_B=E\cap B.
    If both of those are nonempty, then they are separated.
    But E=E_A\cup E_B.
    What is wrong with that?
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  5. #5
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    Re: a strange question

    Quote Originally Posted by Plato View Post
    Let E_A=E\cap A~\&~E_B=E\cap B.
    If both of those are nonempty, then they are separated.
    But E=E_A\cup E_B.
    What is wrong with that?
    are you showing by contridiction? it seems like it would contradicts that A,B are disjoint, but what is separated mean?
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    Re: a strange question

    Quote Originally Posted by wopashui View Post
    are you showing by contridiction? it seems like it would contradicts that A,B are disjoint, but what is separated mean?
    Do you know what it means to say a set is connected?
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  7. #7
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    Re: a strange question

    Quote Originally Posted by Plato View Post
    Do you know what it means to say a set is connected?
    we just defined eonnected set as not disconnected, if we prove by contridiction, what should we suoopse, what is the neglation?
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    Re: a strange question

    Quote Originally Posted by wopashui View Post
    we just defined eonnected set as not disconnected, if we prove by contridiction, what should we suoopse, what is the neglation?
    Connected set as not disconnected.
    That is correct. BUT being disconnected means not being the union of two separated sets.

    The statement that A~\&~B are separated means that each is non-empty and neither contains a point nor a limit point of the other.
    Last edited by Plato; November 24th 2011 at 07:09 AM.
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  9. #9
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    Re: a strange question

    separated means disjoint, right?
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  10. #10
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    Re: a strange question

    Quote Originally Posted by wopashui View Post
    separated means disjoint, right?
    Well it is more that just that, although they are disjoint.
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  11. #11
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    Re: a strange question

    Quote Originally Posted by wopashui View Post
    separated means disjoint, right?
    their closure is disjoint from the other set
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