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Math Help - Metric Space, closed sets in a closed ball

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    Question 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.
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    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.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by hebby View Post
    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 x of radius r as B_r(X,x). Now suppose that \xi was a limit point of B_r(X,x) but not an element of B_r(X,x). Then every open ball around \xi would contain another point of B_r(X,x) besides \xi. Therefore d(x,\xi)\le r+\varepsilon\quad\forall\varepsilon>0. Assume that d(x,\xi)>r\implies d(x,\xi)-r>0, then choosing \varepsilon=d(x,\xi)-r would derive a contradiction. Therefore d(x,\xi)\le r and the conclusion follows.
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