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Math Help - Problem about conected spaces

  1. #1
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    Problem about conected spaces

    Hello, this is a problem from Munkres: Let X and Y be conected spaces such that Y\subset X. If A and B are a separation of X-Y prove that Y\cup A\,and\,Y\cup B are conected.
    I can only prove that Y\cup A\,or\,Y\cup B is conected and since I've read from a Spanish transation I wonder whether the problem is correctly formulated.
    Last edited by facenian; August 13th 2010 at 06:46 AM. Reason: gramma fixing
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  2. #2
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    What if U~\&~V is a separation of A\cup Y?
    We know that because Y is connected we have Y\subseteq U \text{ or } Y\subseteq V.
    Say Y\subseteq U. What does that say about X=U\cup V\cup B?

    Use similar argument for B\cup Y.
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  3. #3
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    I'm sorry but I can not see why U\cup V\cup B should be a separation for  X
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  4. #4
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    Quote Originally Posted by facenian View Post
    I'm sorry but I can not see why U\cup V\cup B should be a separation for  X
    Let C=U\cup B. Then Because Y\subseteq U we have Y\subseteq C.
    Also it must be the case that V\subseteq A.
    But A~\&~B are separated sets.
    What about V~\&~C?
    Is this true V\cup C=X?
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  5. #5
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    You're right, now I see it. Thank you
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