Let D be a nonempty subset of a metric space (X, d ). If x ∈ X, we define the distance from x to D by (a) Prove that d (x, D) = 0 iff x ∈ c1 D. (b) If D is compact, prove that there exists a point p 0...

Let D be a nonempty subset of a metric space (X, d ). If x ∈ X, we define the distance from x to D by

(a) Prove that d (x, D) = 0 iff x ∈ c1 D.

(b) If D is compact, prove that there exists a point p₀
∈ D such that d (x, D) = d (x, p₀).

(c) Prove that f (x) = d (x, D) is a uniformly continuous function on X (even when D is not compact).