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.

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- Nov 19th 2009, 03:19 PMhebbyMetric 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. - Nov 19th 2009, 04:43 PMFocus
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.

- Nov 19th 2009, 05:30 PMDrexel28
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.