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Math Help - Union of Connected Sets

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    Union of Connected Sets

    How do I use proof by contradiction to show that the union of two connected sets is connected?
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    Quote Originally Posted by amoeba View Post
    How do I use proof by contradiction to show that the union of two connected sets is connected?
    The statement as it is now is false.

    Consider A=[0,1] and B=[2,3]. A and B are clearly connected, but A\cup B isn't.

    You would have to assume that A and B are not disjoint, that is, \exists~x such that x\in A and x\in B.
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    Quote Originally Posted by amoeba View Post
    How do I use proof by contradiction to show that the union of two connected sets is connected?
    Note: you need to assume (for instance) that the two connected subsets have a non-empty intersection.

    Let X,Y be connected sets such that X\cap Y\neq\emptyset. Assume by contradiction that X\cup Y is disconnected. Then there would exist two disjoint non-empty open subsets O_1,O_2 of X\cup Y such that O_1\cup O_2=X\cup Y (nb: O_1,O_2 are open for the induced topology on X\cup Y). Let U_1=O_1\cap X, U_2=O_2\cap X. Then U_1,U_2 are disjoint open subsets of X such that U_1\cup U_2=X. Similarly, we can define V_1,V_2 for Y. Since X and Y are connected, we must have either U_1=\emptyset or U_2=\emptyset, and either V_1=\emptyset or V_2=\emptyset. Each case leads to a contradiction. For instance, if U_1=\emptyset and V_1=\emptyset, then U_1\cup V_1=\emptyset, whereas U_1\cup V_1=O_1\neq\emptyset. If U_1=\emptyset and V_2=\emptyset, then X\cap O_1=\emptyset and Y\cap O_2=\emptyset, hence (X\cap Y)\cap(O_1\cup O_2)=\emptyset ( O_1,O_2 are disjoint), which contradicts X\cap Y\neq \emptyset since O_1\cup O_2=X\cup Y.
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    Quote Originally Posted by amoeba View Post
    How do I use proof by contradiction to show that the union of two connected sets is connected?

    you cannot prove that because it isn't true: for example (0,1) \/ (1,2) is the union of two connected sets but it isn't connected

    Tonio
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    Quote Originally Posted by Laurent View Post
    Let U_1=O_1\cap X, U_2=O_2\cap X. Then U_1,U_2 are disjoint open subsets of X such that U_1\cup U_2=X .
    How do you know that U1 and U2 are disjoint?

    Also, you said

    Then U_1,U_2 are disjoint open subsets of X such that U_1\cup U_2=X.

    ==

    How do you know this fact that the union of U1 and U2 is X?
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    Quote Originally Posted by amoeba View Post
    How do you know that U1 and U2 are disjoint?
    Because U_1\subset O_1, U_2\subset O_2, and O_1,O_2 are disjoint.

    Also, you said

    Then U_1,U_2 are disjoint open subsets of X such that U_1\cup U_2=X.

    ==

    How do you know this fact that the union of U1 and U2 is X?
    Because X\subset X\cup Y = O_1\cup O_2. You should draw a diagram if you don't get it. I just split X into two parts according to the partition of X\cup Y. If O_1,O_2 cut X\cup Y in two disjoint pieces, then their intersections with X cut it also in two disjoint pieces.
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    How come you assumed in your post that O1 U O2 = U1 U U2?

    Why isn't showing that the pair S1, S2 isn't a disconnection sufficient?
    Last edited by amoeba; October 16th 2009 at 07:52 PM.
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  8. #8
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    Quote Originally Posted by amoeba View Post
    How come you assumed in your post that O1 U O2 = U1 U U2?
    I did not. In fact, U_1\cup U_2= X, as results directly from the definitions of U_1 and U_2.

    Why isn't showing that the pair S1, S2 isn't a disconnection sufficient?
    Which S_1, S_2 ?? Since you procede by contradiction, you must find a disconnection of either X or Y, this would be the contradiction.
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