Ball and Sphere (metric space)

In a metric space, the ball of a given radius around its centre is the set of points with a distance to the centre less than the radius, where the notion of distance comes from the metric. Balls provide the basis of defining the notion of an open set, and many other fundamental ideas in metric spaces.

Open Ball

Given a metric space \((X, d)\) the open ball of radius \(r\) centred at \(x_{0}\) is defined to be:

\[ B_{r}(x_{0}) = B(r, x_{0}) = \{x \in X : d(x, x_{0}) < r\}.\]

Note specifically the use of the \(<\) sign, which means the set of points exactly \(r\) distance from \(x_0\) are not included in the ball.

Closed Ball

The closed ball is defined in much the same way, with the only difference being that now the points \(r\) distance from \(x_0\) are inlcuded.

Definition

Given a metric space \((X, d)\), the closed ball with radius \(r\) and centre \(x_{0}\) is defined as:

\[ \overline{B_{r}(x_{0})} = \overline{B(r, x_{0})} = \{x \in X : d(x, x_{0}) \leq r\}.\]

Note the use of \(\overline{B(r, x_{0})}\) since a closed ball defined as above is in fact the closure of the corresponding open ball.

Sphere

Definition

Often the boundary of a ball is called a sphere.

For an intuition for how balls change depending on the metric used, see unit balls in \(\mathbb{R}^2\).