Open and closed balls are the local measuring devices of analysis. In a [Metric Space](/page/Metric%20Space), they turn the numerical distance between two points into neighbourhoods, convergence tests, continuity tests, and compactness arguments in finite-dimensional Euclidean spaces. In Euclidean analysis they are the standard regions on which estimates are localized, while in [Topology](/page/Topology) they are the model examples behind [Open Set](/page/Open%20Set), [Closed Set](/page/Closed%20Set), [Closure](/page/Closure), and [Interior](/page/Interior).

A ball answers a simple question: which points lie within a prescribed distance of a centre? The open version records strict nearness and is built for perturbation arguments. The closed version records bounded nearness and is built for limiting arguments. Many basic statements in analysis use both at once: a function may be controlled on an open ball, extended to a closed ball, and then studied through the boundary between them.

## Definition

The most useful neighbourhoods in a metric space are those determined by the metric itself. If $x_0$ is the point of interest and $r>0$ is the scale of observation, the open ball consists of all points that can move a positive amount before reaching distance $r$ from $x_0$.

[definition: Open Ball] Let $(X,d)$ be a metric space, let $x_0 \in X$, and let $r>0$. The open ball of radius $r$ centred at $x_0$ is \begin{align*} B(x_0,r) := \{x \in X : d(x,x_0)<r\}. \end{align*} [/definition]

Limit arguments often produce points only as limits of approximating points. A strict inequality can be lost at the limit, so one needs the version of a ball that permits points exactly at distance $r$ from the centre.

[definition: Closed Ball] Let $(X,d)$ be a metric space, let $x_0 \in X$, and let $r>0$. The closed ball of radius $r$ centred at $x_0$ is \begin{align*} \overline{B}(x_0,r) := \{x \in X : d(x,x_0)\le r\}. \end{align*} [/definition]

In Euclidean estimates, the notation $B(x_0,r)$ is usually used without naming the metric each time. To make that convention unambiguous, we need a separate Euclidean definition in which distance is measured by the norm $|x-x_0|$.

[definition: Euclidean Open Ball] Let $x_0 \in \mathbb{R}^n$ and let $r>0$. The Euclidean open ball of radius $r$ centred at $x_0$ is \begin{align*} B(x_0,r) := \{x \in \mathbb{R}^n : |x-x_0|<r\}. \end{align*} [/definition]

When estimates include limiting points or boundary values, the strict Euclidean inequality is too small. To handle compactness and boundary arguments in Euclidean space, we need the version that keeps all points whose Euclidean distance from the centre is at most the chosen radius.

[definition: Euclidean Closed Ball] Let $x_0 \in \mathbb{R}^n$ and let $r>0$. The Euclidean closed ball of radius $r$ centred at $x_0$ is \begin{align*} \overline{B}(x_0,r) := \{x \in \mathbb{R}^n : |x-x_0|\le r\}. \end{align*} [/definition]

The points added when one passes from the open Euclidean ball to the closed Euclidean ball form the boundary at exactly one fixed distance from the centre. Boundary-value problems and surface integrals need this set as an object in its own right.

[definition: Sphere] Let $x_0 \in \mathbb{R}^n$ and let $r>0$. The sphere of radius $r$ centred at $x_0$ is \begin{align*} \partial B(x_0,r) := \{x \in \mathbb{R}^n : |x-x_0|=r\}. \end{align*} [/definition]

A radius of zero cannot describe an open neighbourhood, but closed-ball arguments sometimes allow radii to decrease to zero. To handle that endpoint case cleanly, we use a separate degenerate closed ball whose only possible point is the centre itself.

[definition: Degenerate Closed Ball] Let $(X,d)$ be a metric space and let $x_0 \in X$. The degenerate closed ball centred at $x_0$ is \begin{align*} \overline{B}(x_0,0) := \{x_0\}. \end{align*} [/definition]

[example: Centre as a Zero-Radius Closed Ball] For any metric space $(X,d)$ and any $x_0\in X$, the condition $d(x,x_0)\le 0$ forces $d(x,x_0)=0$, hence $x=x_0$. Thus the zero-radius closed ball records the centre alone. [/example]

Most analysis uses only radii $r>0$ for open balls, because neighbourhoods must contain room around their centre. Closed balls of radius zero appear naturally when a sequence of closed balls shrinks to a single point.

The real line gives the quickest calibration for the notation. In one dimension, the metric definition recovers the intervals used in limits, so the symbols $B(x_0,r)$ and $\overline{B}(x_0,r)$ should be read as distance-based versions of familiar open and closed intervals.

[example: Basic Real-Line Balls] In $\mathbb{R}$ with the Euclidean metric $d(x,y)=|x-y|$, the open ball centred at $x_0\in\mathbb{R}$ with radius $r>0$ is computed from the inequality $|x-x_0|<r$. For any $x\in\mathbb{R}$, \begin{align*} x\in B(x_0,r) \Longleftrightarrow |x-x_0|<r. \end{align*} By the defining property of absolute value on $\mathbb{R}$, this is equivalent to \begin{align*} -r<x-x_0<r. \end{align*} Adding $x_0$ to all three parts gives \begin{align*} x_0-r<x<x_0+r. \end{align*} Therefore \begin{align*} B(x_0,r)=(x_0-r,x_0+r). \end{align*} The closed ball is obtained by replacing strict inequality with non-strict inequality. For any $x\in\mathbb{R}$, \begin{align*} x\in \overline{B}(x_0,r) \Longleftrightarrow |x-x_0|\le r. \end{align*} Again using the absolute value inequality on $\mathbb{R}$, this is equivalent to \begin{align*} -r\le x-x_0\le r. \end{align*} Adding $x_0$ to all three parts gives \begin{align*} x_0-r\le x\le x_0+r. \end{align*} Hence \begin{align*} \overline{B}(x_0,r)=[x_0-r,x_0+r]. \end{align*} Thus, on the real line, metric open and closed balls are exactly the familiar open and closed intervals centred at $x_0$. [/example]

## Equivalent Characterisations