Thread: Metric Space, closed sets in a closed ball

Let(X, d) be a metric space. the set {y ∈ X : d(x, y) ≤ r} is a closed ball centered at X and with radius r.
(a)Show that a closed ball is a closed set.

Have you tried any of the questions you posted? If so post your working out. A tip to get you started, pick a point outside of the ball and find and open ball that contains it.

Originally Posted by hebby

Let(X, d) be a metric space. the set {y ∈ X : d(x, y) ≤ r} is a closed ball centered at X and with radius r.
(a)Show that a closed ball is a closed set.

Denote the closed ball centered at of radius as . Now suppose that was a limit point of but not an element of . Then every open ball around would contain another point of besides . Therefore . Assume that , then choosing would derive a contradiction. Therefore and the conclusion follows.