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  3. q2. let (e, d) be a metric space, and let x ? e. w

Q2. Let (E, d) be a metric space, and let x ? E. We say...

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Advance Math

Q2. Let (E, d) be a metric space, and let x ? E. We say that xis isolated if the set {x} is open in E.

(a) Suppose that there exists r > 0 such that Br(x) containsonly finitely many points. Prove that x is isolated.

(b) Let E be any set, and define a metric d on E by setting d(x,y) = 0 if x = y, and d(x, y) = 1 if x and y are not equal. Provethat every point x ? E is an isolated point. The metric d is calleda discrete metric, and the space (E, d) is called a discrete metricspace.

(c) Let (E, d) be a discrete metric space, as in part (b). Provethat every subset of E is both open and closed.

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