Open ball, closed ball and sphere in a metric space

Let $(X,d)$ be a metric space (Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $x \in X$ and let $r \in \mathbb{R}$ with $r > 0$ (Order on the reals). Define

$B(x,r) := \{\, y \in X : d(x,y) < r \,\}, \qquad \bar B(x,r) := \{\, y \in X : d(x,y) \le r \,\}, \qquad S(x,r) := \{\, y \in X : d(x,y) = r \,\}.$

$B(x,r)$ is the open ball, $\bar B(x,r)$ the closed ball and $S(x,r)$ the sphere of centre $x$ and radius $r$. The radius is always a strictly positive real; a ball of radius $0$ or of negative radius is never written in this library.

Immediate consequences of the definitions. For every $x \in X$ and $r > 0$:

• $x \in B(x,r)$, because $d(x,x) = 0 < r$ (axiom (M1) of Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric); in particular open and closed balls are nonempty.
• $B(x,r) \subseteq \bar B(x,r)$ and $S(x,r) \subseteq \bar B(x,r)$, and $\bar B(x,r)$ is the disjoint union of $B(x,r)$ and $S(x,r)$, by trichotomy of the order of $\mathbb{R}$ (Complete ordered field (least-upper-bound property), Ordered field): each $y$ satisfies exactly one of $d(x,y) < r$, $d(x,y) = r$, $d(x,y) > r$.
• If $0 < s \le r$ then $B(x,s) \subseteq B(x,r)$ and $\bar B(x,s) \subseteq \bar B(x,r)$, by transitivity of the order.
• Nonnegativity of the metric (Nonnegativity of a metric is a consequence of the other axioms, not an axiom) is what forces the radius convention, and it forces it for the open ball only: if $r \le 0$ then $B(x,r) = \{y : d(x,y) < r\}$ is empty, because $d(x,y) \ge 0 \ge r$ for every $y$. The other two sets behave differently at $r = 0$, and the convention $r > 0$ excludes them for uniformity rather than for emptiness: $\bar B(x,0) = S(x,0) = \{x\}$, since $d(x,y) \le 0$ together with $d(x,y) \ge 0$ gives $d(x,y) = 0$ and hence $y = x$ by (M1). For $r < 0$ all three sets are empty.

A sphere may be empty, and so the three sets are not on a par. For $r > 0$ the open and closed balls always contain $x$, but nothing in the definition produces a point at distance exactly $r$ from $x$. If a metric takes only the values $0$ and $1$, as the discrete metric on the companion page does, then $S(x,2) = \emptyset$ while $B(x,2) = \bar B(x,2)$ is the whole space. So nonemptiness of a sphere is never available by convention: where it is used, it is proved.

The ambient space is part of the notation. $B(x,r)$ depends on $(X,d)$ and not on $x$ and $r$ alone. When more than one space or more than one metric is in play we write $B_X(x,r)$, or $B_d(x,r)$, and likewise for $\bar B$ and $S$. This matters as soon as subspaces appear (Isometry, isometric embedding, and the subspace metric on a subset): a ball of a subspace is the trace on it of a ball of the ambient space, and the two are different sets.

• The names "open ball" and "closed ball" are justified, not merely suggestive. That $B(x,r)$ is an open set and $\bar B(x,r)$ a closed set in the metric topology is proved in Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed; the words are used here only as names for the three sets displayed above.
• The closed ball is not in general the closure of the open ball, and the sphere is not in general the boundary of either. Both failures are recorded on this page as FALSE: in every metric space the closure of $B(x,r)$ is the closed ball of radius $r$ and witnessed on the companion page. The safe reading of the three names is the displayed one and nothing more.