Solution: For the following statement S and proposed proof, either (1) S is true and the

Chapter 16, Problem 16.30

(choose chapter or problem)

For the following statement S and proposed proof, either (1) S is true and the proof iscorrect, (2) S is true and the proof is incorrect, or (3) S is false and the proof is incorrect.Explain which of these occurs.S: Let (X, d) be a metric space. For every open set O in X such that O = , and everyelement b O, there exists an open sphere Sr(b) in X such that Sr(b) and O are disjoint.Proof. Let r = min{d(b, x) : x O}. Consider the open sphere Sr(b). We claim thatSr(b)O = . Assume, to the contrary, that Sr(b)O = . Then there exists y Sr(b)O.Since y Sr(b), it follows that d(b, y) < r. However, since y O, this contradicts the factthat r is the minimum distance between b and an element of O.