MATH527_HW3

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

• Notes
• 2

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern