A punctured ball is an open ball with its centre removed. In the real line it has two connected components. In Euclidean space of dimension it is path-connected: for any two points, choose a third point inside the ball outside the two lines through the centre and those points. Such a point exists since two lines do not cover an open ball in dimension at least two. The two line segments joining through that point stay inside the ball, which is a convex set, and avoid its centre. They form a continuous path.
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