Let \((M,d)\) be a metric space . For a given \(x\in M\) and \(r>0\) the (open) ball of radius \(r\) around \(x\) is the set

\begin{equation*} B_r(x)=\{y\in M\mid d(x,y) The closed ball of radius \(r\) around \(x\) is defined by

\begin{equation*} \bar{B}_r(x)=\{y\in M\mid d(x,y)\le r\}. \end{equation*}

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• open set