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Question 3 (a) (i) Let C be a subset of a metric space. Use sequences to state what it means to say that C is a compact set. (ii) Let C be a nonempty compact set in the metric space (IR, . I) and let p E R. Show that the set given by A={x+p|xEC}, = is compact. Note: It is of no importance whether p is an element of C. (b) Let (X, d) be a metric space and 0 # C C X. (i) Let u = (Uili E , 1} be a family of subsets of X. What does it mean to say that u is an open cover for C? What does it mean to say that U has a finite subcover for C? (ii) Use open covers to state what it means to say that C is compact. (iii) Assume that C is a compact set and let p € X \ C. Show, by means of construction, that there are two disjoint open sets E1 and E2 such that p € E1 and C C E2. You could support your proof with a graphical illustration. Hint: Recall that if X # y then there is € > 0 such that Vc(x)nve(y) = 0 and use the open cover definition of compactness. (c) Let (X, d) be a metric space. (i) What does it mean to say that X is complete? (ii) We want to show that if X is complete and E C X is closed, then E is complete. For this, the following proof is suggested: Let (xn) be a sequence in X. Then since X is complete we have that (xn) is convergent. Since E is closed we must have that lim Xn = X € E and so E no+ is complete. This proof is wrong. Answer briefly the following two questions and then give a correct proof. (q1) Shall we start the proof by taking a sequence (xn) in X? If not, then how shall we start? (q2) Given that (xn) is a convergent sequence in X does it make sense to say that since E is closed then lim Xn = X € E? Explain. n->oo

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