Math 322 { Mathematics Homework 6 Due Oct 19 2010 Solve the following problems. Show all your work. 1. Show that the set frp2 + p3 : r 2 Qg is dense in R 2. For A;B R dene A + B = fx + y : x 2 A; y 2 Bg Prove that if A and B are bounded below then so is A + B and in that case inf(A + B) = infA + infB. 3. Consider the set R2 as a metric space with the distance between x = (x1; x2)andy = (y1; y2) dened as (i) d1(x; y) = maxfjx1 ?? y1j; jx2 ?? y2jg and (ii)d2(x; y) = p(x1 ?? y1)2 + (x2 ?? y2)2 Show that a set A R2 is open (closed) in one of these spaces if and only if it is open (closed) in the other. 4. (a) Show that if A and B are open dense subsets of a metric space X then A B is also dense in X. (b) What happens if we drop the requirement of A and B being open? 5. Let X be a metric space and A be its subset. Prove that the complement of the interior of A is the closure of the complement of A. 6. Let A = f12 ; 23 ; 34 ; :::; n??1 n ; :::g and B = A f1g (a) Show directly that every open cover H of B admits a nite subcover. (b) Give an example of an open cover K of A which does not admit a nite subcover. 1
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