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Math Help - Disjoint connected

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    Disjoint connected

    Suppose  A,B are connected subsets of some metric space  X . Prove that if  A \cap B \neq \emptyset the  A \cup B is connected.

    So suppose  A \cup B is not connected.

    Then  A \cup B = C \cup D where  C and  D are disjoint and clopen. So  C = E_{O} \cap X and  C  = E_{C} \cap X . Also  D = F_{O} \cap X and  D = F_{C} \cap X .

    So  A \cup B = (E_{O} \cap X) \cup (F_{O} \cap X) . Now what? How do you conclude that  A \cap B = \emptyset ?
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    Quote Originally Posted by heathrowjohnny View Post
    Suppose  A,B are connected subsets of some metric space  X . Prove that if  A \cap B \neq \emptyset the  A \cup B is connected.
    So suppose  A \cup B is not connected.
    Then  A \cup B = C \cup D where  C and  D are disjoint and clopen.
    At that point, show that A must be a subset of either C or D.
    Then B must be a subset of the other. There is your contradiction.
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    Quote Originally Posted by Plato View Post
    At that point, show that A must be a subset of either C or D.
    Then B must be a subset of the other. There is your contradiction.
    Isn't this also contropostition? But this all depends on relative openess/closeness right?

    Writing  C = E_{O} \cap X is what you should do? E.g.  E_O is an open set set. We have to show  C is open relative to  A \cup B right?

    So want to show that  A \subseteq E_{O} \cap X and so on....?
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    Quote Originally Posted by heathrowjohnny View Post
    We have to show  C is open relative to  A \cup B right?
    I have no idea how your textbook/instructor requires you to prove connectivity.
    The usual definition simply says that X is connected if and only if there do no exist two sets C & D such that \begin{gathered}  X \cap C \ne \emptyset \;,\;X \cap D \ne \emptyset \;\& \;X \subseteq C \cup D \hfill \\  \overline C  \cap D = \emptyset \;\& \;\overline D  \cap C = \emptyset  \hfill \\ \end{gathered} .
    Put simply: X is not a subset of the union of two sets, each having a nonempty intersection with X, and neither contains a point or a limit point of the other. That is known as a separation of X.
    It seems that you are to use some other approach to connectivity.
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