Suppose K is a nonempty compact subset of a metric space X and x ∈ X....

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 distinct points 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 show there 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 ∈ X with no closest point in K? Is K closed, bounded, complete, compact?

(iv) Let E = {e0, e1, . . . } be a countable set. Define a metric d on E by d(ej , ek) = 1 for j not equal k and j, k not equal 0; d(ej , ej ) = 0 and d(e0, ej ) = 1 + 1/j for j not equal 0. Show d is a metric on E. Let K = {e1, e2, . . . } and x = e0. Is there a closest point in K to x? Is K closed, bounded, complete, compact?

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Expert Solution

Colby Messinger answered 1 year ago

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