Remark

We can have $d(A, B) = 0$ for two subsets $A, B$ of $X$ such that $\overline{A} \cap \overline{B} = \varnothing$, provided that neither subset consists of a single point. For example, on the real line $\mathbf{R}$, the set of integers $> 0$ and the set of points of the sequence $(n + 1/2^n)_{n \geq 1}$ are two disjoint closed sets whose distance a part is zero.

However, if $A$ is compact and $B$ is closed, the relation $d(A, B) = 0$ implies $A \cap B \neq \varnothing$, for by virtue of the relation

it follows from Proposition 3 and Weierstrass's theorem that there exists $x_0 \in A$ such that $d(x_0, B) = d(A, B) = 0$ and hence (Proposition 2) $x_0 \in B$.

The diameter of a non-empty subset $A$ of $X$ is the number (finite or equal to $+\infty$)

The notion of a "W_a-small set" (Chapter II, § 3, no. 1) is identical with that of a set of diameter $\leq a$. A non-empty set $A$ consists of a single point if and only if $\delta(A) = 0$.

A subset $A$ of $X$ is bounded (with respect to the metric $d$) if its diameter is finite; equivalently, if for each point $x_0 \in X$, $A$ is contained in a ball with centre $x_0$. Every subset of a bounded set is bounded, and the union of a finite family of bounded sets is a bounded set.

Note that a subset of $X$ can be bounded with respect to a metric $d$ but unbounded with respect to a metric equivalent to $d$ (cf. § 1, no. 2).