1. In a metric space (X, d ) the closed ball of radius ε > 0 about the point x in X is the setB (x; ε) = {y ∈ X : d (x, y) ≤ ε }.(a) Prove that B (x; ε ) is a closed set.(b) Prove that cl N (x; ε ) ⊆ B (x; ε ).(c) Find an example of a metric space (X, d ), a point x ∈ X, and a radius ε > 0 such that cl N (x; ε ) ≠ B (x; ε ).2. Let D be a subset of a metric space (X, d ).(a) Prove that D is dense in X iff every nonempty open subset of X has a nonempty intersection with D.(b) Prove that D is dense in X iff for every x ∈ X and every ε > 0 there exists a point z in D such that d (x, z)

1. In a metric space (X, d ) the closed ball of radius ε > 0 about the point x in X is the set B (x; ε) = {y ∈ X : d (x, y) ≤ ε }. (a) Prove that B (x; ε ) is a closed set. (b) Prove that cl N (x; ε )...

1. In a metric space (X, d ) the closed ball of radius ε > 0 about the point x in X is the set

B (x; ε) = {y ∈ X : d (x, y) ≤ ε }.

(a) Prove that B (x; ε ) is a closed set.

(b) Prove that cl N (x; ε ) ⊆ B (x; ε ).

2. Let D be a subset of a metric space (X, d ).

(a) Prove that D is dense in X iff every nonempty open subset of X has a nonempty intersection with D.

(b) Prove that D is dense in X iff for every x ∈ X and every ε > 0 there exists a point z in D such that d (x, z)

May 05, 2022