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

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

(a) Suppose that there exists r > 0 such that Br(x) contains only 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. Prove that every point x ∈ E is an isolated point. The metric d is called a discrete metric, and the space (E, d) is called a discrete metric space.

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

Expert Solution

Colby Messinger answered 2 years ago

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

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