MATH527_HW3 - Serge Ballif MATH 527 Homework 3 Problem 1...

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Serge Ballif MATH 527 Homework 3 September 21, 2007 Problem 1. Let X be a compact Hausdorff space. Let K 1 K 2 . . . be a sequence of closed connected subsets of X . Let K = i =1 K i . Show that K is connected. We note first that X is normal since it is compact Hausdorff. Also, K and each set K i are closed as an intersection of closed sets. Furthermore, K is normal as a closed subspace of a normal space. Seeking a contradiction we assume that there is a separation of K consisting of closed sets A and B ( A B = K and A B = ). By the Urysohn lemma there exists a continuous function f : K [0 , 1] such that f ( x ) = 0 for every x in A and f ( x ) = 1 for every x in B . By the Tietze extension theorem the map f can be extended to a continuous map ¯ f : X [0 , 1]. We now consider the sets ¯ f ( K i ) [0 , 1]. Each set ¯ f ( K i ) is connected as a continuous image of a connected set. Furthermore, since each set K i contains the sets A and B , we know that ¯ f ( K i ) contains the points f ( A ) = 0 and f ( B ) = 1. This means that ¯ f ( K i ) = [0 , 1] for all i . Thus for each i there is an element k K i such that f ( k ) = . 5. Therefore, k K . This contradicts the fact that
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