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Math Help - closed subsets of a connected space X

  1. #1
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    closed subsets of a connected space X

    Let A\& B be closed subsets of a connected space X such that X = A \cup B.
    How to show A\& B are connected, if A \cap B is connected.

    I traied this:
    Suppose B is not connected and let B = S \cup T be sepration of B, then,
    A \cap B must lie in S (say) So,
    X = \left( {A \cup S} \right) \cup T
    form separation of x
     \Rightarrow  \Leftarrow

    is this right
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  2. #2
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    Re: closed subsets of a connected space X

    Quote Originally Posted by rqeeb View Post
    Let A\& B be closed subsets of a connected space X such that X = A \cup B.
    How to show A\& B are connected, if A \cap B is connected. I traied this:
    Suppose B is not connected and let B = S \cup T be sepration of B, then,
    A \cap B must lie in S (say) So,
    X = \left( {A \cup S} \right) \cup T
    form separation of x
     \Rightarrow  \Leftarrow
    Now say that likewise A must be connected.
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  3. #3
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    Re: closed subsets of a connected space X

    Quote Originally Posted by Plato View Post
    Now say that likewise A must be connected.
    but now the problem is: To show AUS and S are spen in X
    we know that S & T are open in B, and B is closed in X. Bur how to show they r open in X?
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