What is metric space?

Example 4.2

$\R$ with $\abs{x - y}$; $\R^n$ with any of

the set $C(\intcc{a}{b})$ of continuous functions with the sup distance $d_\infty(f, g) = \sup_{\intcc{a}{b}} \abs{f - g}$ (finite: $f - g$ is bounded); any set with the discrete distance ($d(x,y) = 1$ for $x \neq y$). Distances coming from norms are the subject of Chapter 5.

Example 4.4 (Interior, closure, boundary on one set)

In $\R$, let $A = \intoc{0}{1} \cup \{2\}$. Interior: $\intoo{0}{1}$ — around any $x \in \intoo01$ a small ball stays in $A$; around $1$, every ball $\intoo{1-r}{1+r}$ leaks out of $A$ on the right, so $1$ is not interior; and the isolated $2$ is not interior either. Closure: $\intcc{0}{1} \cup \{2\}$ (the point $0$ is a limit of $A$, nothing else is added). Boundary (closure minus interior): $\{0, 1, 2\}$. Note the asymmetries worth remembering: an endpoint can belong to a set without being interior ($1$), can be adherent without belonging ($0$), and an isolated point is its own boundary ($2$). The same bookkeeping runs verbatim in any metric space, with balls in place of intervals.

Example 4.19 (Distances between sets: compactness earns its keep)

Let $K$ be compact, $F$ closed, $K \cap F = \emptyset$ in a metric space. Then

the function $x \mapsto d(x, F)$ is continuous (Exercise 4.11) and positive on $K$ ($d(x, F) = 0$ would put $x \in \overline F = F$), so it attains a positive minimum on the compact $K$ (Theorem 4.16 (3)). Compactness is not decorative: for two closed sets the infimum can vanish without being attained — in $\R^2$, the hyperbola $F_1 = \{xy = 1\}$ and the axis $F_2 = \{y = 0\}$ are disjoint closed sets with $d(F_1, F_2) = 0$ (the points $(n, \frac1n)$ approach the axis). Escape to infinity is exactly what compactness forbids.