Suppose K is a nonempty compact subset of a metric space X and x? X.
(i) Give an example of an x ? X for which there exists distinctpoints p, r ? K such that, for all q ? K, d(p, x) = d(r, x) ? d(q,x).
(ii) Show, there is a point p ? K such that, for all other q ?K, d(p, x) ? d(q, x).
[Suggestion: As a start, let S = {d(x, y) : y ? K} and showthere is a sequence (qn) from K such that the numerical sequence(d(x, qn)) converges to inf(S).] 63
(iii) Let X = R {0} and K = (0, 1]. Is there a point x ? Xwith no closest point in K? Is K closed, bounded, complete,compact?
(iv) Let E = {e0, e1, . . . } be a countable set. Define ametric d on E by d(ej , ek) = 1 for j not equal k and j, k notequal 0; d(ej , ej ) = 0 and d(e0, ej ) = 1 + 1/j for j not equal0. Show d is a metric on E. Let K = {e1, e2, . . . } and x = e0. Isthere a closest point in K to x? Is K closed, bounded, complete,compact?
