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

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