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Unformatted text preview: CONNECTED COMPONENTS Recall the definition of connectedness (2.45 in ). Definition 1. Let ( X,d ) be a metric space. A subset E X is called disconnected (or sepa- rated ) if there exists nonempty A,B E such that E = A B , and A B = A B = . A subset is called connected if it is not disconnected. Example 2. In R with the Euclidean metric, consider the set E = [0 , 1] (2 , 3) . Since (2 , 3) = [2 , 3] , the strong condition [0 , 1] (2 , 3) = holds, which implies that the two subsets A = [0 , 1] and B = (2 , 3) form a separation of E . Hence E is not connected. Example 3. Again in R , consider the set E = [0 , 1) (1 , 2] . This set is also disconnected: setting A = [0 , 1) and B = (1 , 2] we have A = [0 , 1] which does not intersect (1 , 2] = B , while (1 , 2] = [1 , 2] does not intersect [0 , 1) = A . Thus, E is disconnected. In this case, A B is not empty (it contains the point 1 ). We might say that A and B are adjacent . But they are not connected. A person living inthey are not connected....
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- Spring '09