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Open Ball**

Given a metric space \((X,d)\), a point \(x\in X\) and a radius \(r\in\mathbb{R}_{>0}\), the open ball about \(x\) of radius \(r\) is the subset of all points of \(X\) closer to \(x\) than \(r\); \(B_r(x):=\{y\in X|d(x,y)\lt r\}\).