Open Ball

An open ball in a metric space is the set of all points whose distance from a given center is strictly less than a specified positive radius. It generalizes an open interval on the real line and the interior of a circular disk or solid sphere in Euclidean space. Open balls connect distance with topology: they provide the basic sets from which the topology induced by a metric is constructed. Their shape depends on the metric, not merely on the underlying set. (math.utoronto.ca)

Definition and notation

Let $(X,d)$ be a metric space, let $a\in X$, and let $r>0$ be a real number. The open ball with center $a$ and radius $r$ is

Common alternatives include $B(a,r)$ and $B_r(a)$. The subscript identifying the metric is useful when several metrics are under consideration. The strict inequality excludes every point at distance exactly $r$, while the center belongs to the ball because $d(a,a)=0$. (jirka.org)

The radius is conventionally positive. If the same formula is extended to $r=0$, it gives the empty set, not the singleton containing the center. For a fixed center, balls are nested: $0<r\le s$ implies $B(a,r)\subseteq B(a,s)$. These statements follow directly from the defining inequality. (sites.millersville.edu)

Euclidean examples

On the real line with distance $d(x,y)=|x-y|$,

In $\mathbb R^n$, using Euclidean distance,

Thus a two-dimensional ball is an open disk, and a three-dimensional ball is the interior of a solid sphere. A ball includes interior points; it is not merely the surrounding circumference or spherical surface. (math.utoronto.ca)

Openness is relative to the ambient space. If $Y$ is a subset of $X$, equipped with the restricted metric, then

For example, in $Y=[0,1]$ with ordinary distance, $B_Y(0,\tfrac12)=[0,\tfrac12)$. This is open in $Y$, although not in $\mathbb R$. Such examples express the subspace topology. (jirka.org)

Why balls are open

Every open ball is an open set in the topology induced by its metric. The key argument uses the triangle inequality. If $y\in B(a,r)$, choose

Whenever $z\in B(y,\delta)$,

Consequently $B(y,\delta)\subseteq B(a,r)$: every point of the original ball has a smaller ball around it that stays inside. (jirka.org)

More generally, $U\subseteq X$ is open precisely when every $x\in U$ has some $r>0$ satisfying $B(x,r)\subseteq U$. Equivalently, open sets are arbitrary unions of open balls. The collection of all balls therefore forms a basis for the metric topology and makes $X$ a topological space. Finite intersections of balls are open, though they need not themselves be balls. (cs.mcgill.ca)

Closed balls and closure

The corresponding closed ball is

It is a closed set. However, the notation must not be confused with the closure of the open ball: in a general metric space,

and equality can fail. In a normed vector space, equality holds for every positive radius. (homepages.ucl.ac.uk)

For a concrete counterexample, give a set $X$ with at least two points the discrete metric, where distinct points have distance $1$. Then $B(a,1)=\{a\}$, whose closure is itself, whereas $\overline B(a,1)=X$. More generally, its open balls are singletons for $0<r\le1$ and the whole space for $r>1$. Thus “open” does not necessarily mean “not closed.” (jirka.org)

Norms and geometric shape

In a normed vector space, the distance is $d(x,y)=\|x-y\|$, where $\|\cdot\|$ is a norm. Hence

Every positive-radius ball is a translation and scaling of the open unit ball $B(0,1)$, and every norm ball is a convex set. Convexity follows from the norm’s triangle inequality and homogeneity. (hhhyx817.github.io)

Different norms produce different shapes. In $\mathbb R^2$, the Euclidean norm gives circular disks; $\|(x,y)\|_1=|x|+|y|$ gives diamond-shaped balls; and $\|(x,y)\|_\infty=\max(|x|,|y|)$ gives squares with their edges excluded. These shapes illustrate why “ball” is a distance-based term rather than a requirement of visual roundness. (jirka.org)

Roles in analysis

Open balls provide a precise language for neighborhoods and limits. A sequence converges to $a$ exactly when it eventually lies in every $B(a,r)$. A continuous function $f:X\to Y$ is continuous at $a$ exactly when, for every $\varepsilon>0$, some $\delta>0$ satisfies

This is the ball formulation of epsilon–delta continuity. (sites.millersville.edu)

Balls also describe total boundedness: a set is totally bounded when, for every positive radius, finitely many balls of that radius cover it. Within a complete metric space, a subset is compact exactly when it is closed and totally bounded. Open balls themselves need not be compact; the real interval $(0,1)$ is a basic example. (jirka.org)