hw7 - IE 500 Fall 2009 E. Alper Yldrm HOMEWORK 7 (due...

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IE 500 – Fall 2009 E. Alper Yıldırım HOMEWORK 7 (due Monday, December 28 in class) 1. Let ( X,d ) be a metric space and let A X . (a) Prove that (int( A )) C = cl( A C ), where the complement is taken with respect to X . (b) Prove that ∂A = ∂A C . 2. Let ( X,d ) be a metric space and let A n X for each n N . (a) Prove that cl( S k n =1 A n ) = S k n =1 cl( A n ) for each k N . (b) Prove that cl( S n N A n ) S n N cl( A n ). (c) Show, by an example, that the inclusion in part (b) can be proper. 3. Let X be a nonempty set and let d 1 and d 2 be two metrics on X such that there exist positive real numbers 0 < α < β such that αd 2 ( x,y ) d ( x,y ) βd 2 ( x,y ) for all x,y X . (a) Let A
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