Open ball

In a metric space, the open ball of radius $r>0$ about $x$ is $\{y:d(x,y)<r\}$; it is the open-boundary convention for a ball.

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 $n\ge 2$ 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.