Bounded subset, diameter, distance from a point to a set, and distance between two sets 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) and let $A,B\subseteq X$.

Bounded subset. $A$ is bounded if $A=\mathrm{\varnothing }$ or there are $x_0\in X$ and a real $r>0$ with $A\subseteq B(x_0,r)$ (Open ball, closed ball and sphere in a metric space). The space $(X,d)$ is a bounded metric space if $X$ is a bounded subset of itself.

Diameter, for nonempty bounded $A$ only. Suppose $A$ is nonempty and bounded, and put

$D(A):=\{d(a,b):a,b\in A\}\subseteq \mathbb{R}\mathrm{.}$

Then $D(A)$ is nonempty, since $A$ is, and it is bounded above: fixing $x_0$ and $r$ with $A\subseteq B(x_0,r)$, every $a,b\in A$ satisfy $d(a,b)\le d(a,x_0)+d(x_0,b)<r+r$ by the triangle inequality, symmetry (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric) and addition of inequalities (Order is preserved by adding a constant and by adding inequalities, Ordered field), so $r+r$ is an upper bound of $D(A)$ (Lower bound, bounded below, bounded set). Hence $D(A)$ has a least upper bound in $\mathbb{R}$ by the least-upper-bound property (Complete ordered field (least-upper-bound property)), and that bound is unique (Suprema and infima are unique). Define

$\mathrm{diam}(A):=\sup D(A)\mathrm{.}$

Distance from a point to a set, for nonempty $A$ only. Let $x\in X$ and let $A$ be nonempty, and put $E(x,A):=\{d(x,a):a\in A\}$. Then $E(x,A)$ is nonempty and bounded below by $0$, since a metric is nonnegative (Nonnegativity of a metric is a consequence of the other axioms, not an axiom, Lower bound, bounded below, bounded set), so it has a greatest lower bound (Every nonempty set bounded below has an infimum, Greatest lower bound (infimum)), unique by Suprema and infima are unique. Define

$d(x,A):=\inf E(x,A)\mathrm{.}$

Distance between two sets, for nonempty $A$ and $B$ only. Put $E(A,B):=\{d(a,b):a\in A,b\in B\}$, again nonempty and bounded below by $0$, and define

$d(A,B):=\inf E(A,B)\mathrm{.}$

Every one of the three scope restrictions is load bearing. In this library $\sup$ and $\inf$ denote real numbers and are written only after existence has been established; the least-upper-bound property of Complete ordered field (least-upper-bound property) and the greatest-lower-bound property of Every nonempty set bounded below has an infimum apply to nonempty real sets with the respective bounds. Empty or unbounded sets acquire no real supremum or infimum by convention. Accordingly:

• $\mathrm{diam}(A)$ is defined exactly when $A$ is nonempty and bounded. It is not defined for $A=\mathrm{\varnothing }$, and it is not defined, not even as an infinite value, for an unbounded $A$.
• $d(x,A)$ is defined exactly when $A\mathrm{\ne }\mathrm{\varnothing }$, and $d(A,B)$ exactly when both $A$ and $B$ are nonempty. No boundedness is needed for these two, because $0$ is always a lower bound.

Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric

Open ball, closed ball and sphere in a metric space

Order is preserved by adding a constant and by adding inequalities

Ordered field

Lower bound, bounded below, bounded set

Complete ordered field (least-upper-bound property)

Suprema and infima are unique

Nonnegativity of a metric is a consequence of the other axioms, not an axiom

Every nonempty set bounded below has an infimum

Greatest lower bound (infimum)

• Diameter and the distance functions are nonnegative. For nonempty bounded $A$ and any $a\in A$ we have $0=d(a,a)\in D(A)$, so $\mathrm{diam}(A)\ge 0$; and $d(x,A)\ge 0$, $d(A,B)\ge 0$ because $0$ is a lower bound of the sets they are infima of (Nonnegativity of a metric is a consequence of the other axioms, not an axiom, Greatest lower bound (infimum)).
• $d(A,B)$ is not a metric on the nonempty subsets of $X$. It is symmetric and vanishes on $A=B$, but two distinct disjoint sets can be at distance $0$, so the separation axiom (M1) of Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric fails; the witness is on the companion page. The letter $d$ is reused for three different functions here, the metric, the point-to-set distance and the set-to-set distance, only because the arguments make the intended one unambiguous.
• $d(x,A)$ is the special case $d(\{x\},A)$, since $E(\{x\},A)=E(x,A)$, and the two infima therefore agree by uniqueness (Suprema and infima are unique).