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Math Help - Conectedness problem

  1. #1
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    Conectedness problem

    Let X be a connected topological space and A \subset X connected. Suppose that A^c = M \cup N with M and N separated (i.e. cl(M) \cap N = \emptyset and cl(N) \cap M = \emptyset ). Prove that A \cup M and A \cup N are connected.

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  2. #2
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    Quote Originally Posted by Inti View Post
    Let X be a connected topological space and A \subset X connected. Suppose that A^c = M \cup N with M and N separated (i.e. cl(M) \cap N = \emptyset and cl(N) \cap M = \emptyset ). Prove that A \cup M and A \cup N are connected.
    First note that A \cap M = \emptyset \;\& \;A \cap N = \emptyset . WHY?
    And A \cup M \cup N = X and X is connected.
    If either A \cup M or A \cup N were not connected what about X?
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