Closed Metric Ball¶

Definition¶

For a metric space $(X,d)$ , a point $x_0$ in $X$ , and $r > 0$ $0$ , the closed metric ball $\overline{B}_{(X,d)}(x_0,r)$ in $X$ , centred at $x_0$ with radius $r$ in metric $d$ , is defined as the following set:

\overline{B}_{(X,d)}(x_0,r) = \{x \in X \colon d(x, x_0) \leq r\}.

The ball is a subset of the ambient space $X$ : its members are selected by the distance condition, not by adding new points to $X$ .

Interpretation¶

A closed metric ball is the set of all points whose distance from the centre $x_0$ is at most the radius $r$ : the open ball together with the points at distance exactly $r$ . In $\mathbb{R}$ with $d(x,y)=|x-y|$ it is the closed interval $[x_0-r,x_0+r]$ $r]$ .

A disc around x0 with radius r drawn as an arrow; a point on the rim is marked as included, since the closed ball contains all points at distance at most r.

The disc is drawn with a solid rim. The green point on the rim has $d(\boldsymbol{x},\boldsymbol{x}_0)=r$ and is included, because the definition uses $\le$ . In the Euclidean plane the rim is the boundary of the disc, and containing it is what makes the closed ball a closed set.

The closed ball also contains its sphere $\{d(\boldsymbol x,\boldsymbol x_0)=r\}$ $r}$ . Drag $\boldsymbol x$ onto the boundary to see the difference.

Closedness holds in every metric space, by the triangle inequality. If $d(y,x_0)>r$ $r$ , put $s=d(y,x_0)-r>0$ $0$ ; every $z$ with $d(z,y)<s$ has $d(z,x_0)\ge d(y,x_0)-d(y,z)>r$ $r$ . So the complement of $\overline{B}(x_0,r)$ is open.

A common slip comes from the notation: $\overline{B}(x_0,r)$ looks like the closure of $B(x_0,r)$ . The closure is always contained in the closed ball, and in $\mathbb{R}^n$ with a $p$ -metric the two coincide, but in general they differ. With the discrete metric, $B(x,1)=\{x\}$ is already closed, while $\overline{B}(x,1)$ is the whole space.

This idea returns later: closed balls in $\mathbb{R}^n$ are standard examples of compact sets, on which a continuous real-valued function attains a largest and a smallest value.

Key Results¶

The closed metric ball is a closed set.
In a normed vector space, in particular in $\mathbb{R}^n$ with any $p$ -metric, the closed metric ball is the closure of the open metric ball with the same centre and radius. This can fail for a metric that is merely topologically equivalent to these, such as $\min(|x-y|,1)$ on $\mathbb{R}$ with radius $1$ .
In a general metric space, the closed ball $\overline{B}_{(X,d)}(x_0,\varepsilon)$ need not equal $\overline{B_{(X,d)}(x_0,\varepsilon)}$ , the closure of the open ball.

Examples¶

In $\mathbb{R}^2$ with the usual distance, $\overline{B}((0,0),1)$ is the closed disc of radius $1$ centred at the origin. Points with $x^2 + y^2 \leq 1$ are elements of this ball.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 6

Aliases: closed ball Symbols: element of, epsilon, real numbers

Last updated: 20 October 2026 03:02