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Thread: need an example here

  1. #1
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    need an example here

    Let A and B be closed subsets of M, and supposed that both $\displaystyle A \bigcup B$ and $\displaystyle A \bigcap B$ are connected. Prove that A and B are connected.

    Find an example of disconnected subsets A,B of R such that $\displaystyle A \bigcup B $and $\displaystyle A \bigcap B$ are connected.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: need an example here

    Quote Originally Posted by wopashui View Post
    Let A and B be closed subsets of M, and supposed that both $\displaystyle A \bigcup B$ and $\displaystyle A \bigcap B$ are connected. Prove that A and B are connected.

    Find an example of disconnected subsets A,B of R such that $\displaystyle A \bigcup B $and $\displaystyle A \bigcap B$ are connected.
    Ok, I helped you with the other two. What are your ideas for this one?
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    Re: need an example here

    Quote Originally Posted by Drexel28 View Post
    Ok, I helped you with the other two. What are your ideas for this one?
    sorry, I have proved the statement, but i'm stuck with the example part, need help on it
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    Re: need an example here

    Quote Originally Posted by wopashui View Post
    sorry, I have proved the statement, but i'm stuck with the example part, need help on it
    $\displaystyle \begin{gathered} A = (1,2) \cup (3,4) \hfill \\ B = (0,3] \cup [4,6] \hfill \\ \end{gathered} $
    Last edited by Plato; Nov 29th 2011 at 06:58 AM.
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    Re: need an example here

    Quote Originally Posted by Plato View Post
    $\displaystyle \begin{gathered} A = (1,2) \cup (3,4) \hfill \\ B = (0,3] \cup [4,6] \hfill \\ \end{gathered} $
    thanks, in this example, AUB= (0,6) which is connected, and A intersect B= empty set which is connected, and A is disconnected. But is B disconnected? what is the disconnection might be?
    Last edited by Plato; Nov 29th 2011 at 06:59 AM.
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  6. #6
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    Re: need an example here

    Quote Originally Posted by wopashui View Post
    thanks, in this example, AUB= (1,7) which is connected, and A intersect B= empty set which is connected, and A is disconnected. But is B disconnected? what is the disconnection might be?
    ok
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