Metric Balls

Why

We speak of a set of elements of a metric space which are all within some distance of a fixed point.

Definition

The inspiration is the notion of a solid ball in three-dimensional space.

Consider a metric space and an element of the base set. The metric ball of radius $r$ centered at the element is the set of all elements which are less than $r$-distance from the element.

Notation

Let $(A, d)$ be a metric space. Let $a \in A$. Let $r$ be a real number. Then the ball centered at $a$ of radius $r$ is

\[ \Set*{b \in A}{d(a, b) < r}. \]