connect1 - CONNECTED COMPONENTS Recall the definition of...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CONNECTED COMPONENTS Recall the definition of connectedness (2.45 in [1]). 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....
View Full Document

Ask a homework question - tutors are online